
Citation 
 Permanent Link:
 https://ufdc.ufl.edu/UFE0021221/00001
Material Information
 Title:
 Ab Initio Density Functional Theory for OpenShell Systems, Excited States and Response Properties
 Creator:
 Bokhan, Denis
 Place of Publication:
 [Gainesville, Fla.]
Florida
 Publisher:
 University of Florida
 Publication Date:
 2007
 Language:
 english
 Physical Description:
 1 online resource (135 p.)
Thesis/Dissertation Information
 Degree:
 Doctorate ( Ph.D.)
 Degree Grantor:
 University of Florida
 Degree Disciplines:
 Chemistry
 Committee Chair:
 Bartlett, Rodney J.
 Committee Members:
 Monkhorst, Hendrik J.
Hirata, So Ohrn, Nils Y. Hershfield, Selman P.
 Graduation Date:
 12/14/2007
Subjects
 Subjects / Keywords:
 Approximation ( jstor )
Atoms ( jstor ) Density functional theory ( jstor ) Diagrams ( jstor ) Electrons ( jstor ) Energy ( jstor ) Flux density ( jstor ) Orbitals ( jstor ) Perturbation theory ( jstor ) Wave functions ( jstor ) Chemistry  Dissertations, Academic  UF density, effective, excited, functional, hyperpolarizabilities, open, optimized, potential, properties, response, shell, states, system, theory
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) borndigital ( sobekcm ) Electronic Thesis or Dissertation Chemistry thesis, Ph.D.
Notes
 Abstract:
 Ab initio density functional theory (DFT) based on the optimized effective potential (OEP) method is a new approach to study the electronic structure of atomic, molecular and^M solid state systems. It contains elements of both wave function and density functional theories and is free from limitations of conventional DFT because of using orbitaldependent functionals derived from systematic approximations of the wave function theory. Ab initio DFT methods with exchangecorrelation functionals based on manybody perturbation theory (MBPT) have been derived and implemented recently. The exchangecorrelation potentials derived from MBPT have a complicated structure and their derivation in higherorder of MBPT by the use of the chainrule for functional differentiation requires significant effort. To facilitate such derivations, I developed a special diagrammatic formalism for taking functional derivatives. An alternative way to construct OEP MBPT exchangecorrelation potentials is to use the density condition. It makes it possible to obtain potentials for different partitionings of the full molecular Hamiltonian with MBPT. Using the diagrammatic formalism developed for taking functional derivatives, we show an orderbyorder equivalence between the functional derivative and the density condition approaches to OEP MBPT for the case of the KohnSham partitioning of the molecular Hamiltonian. For any other partitionings, different exchangecorrelation potentials are produced by the functional derivative and density condition approaches. The timedependent extension of OEP in the adiabatic approximation with exchangeonly potentials and kernels was recently implemented and applied to some molecular systems. The corresponding excitation energies and polarizabilities are in good agreement with timedependent HartreeFock results. However, such results cannot be used for comparison with experimental values due to lack of a description of electron correlation effects. In order to account for correlation effects, a correlation kernel corresponding to the MBPT(2) potentials has been derived. Its structure and properties are described in detail. The first numerical results for excitation energies with pure ab initio potentials and kernels are presented. The recently implemented OEP MBPT(2) method has been extended to the case of spinpolarized openshell systems. The total energies obtained for several openshell systems are very close to the corresponding values obtained with the highlycorrelated coupled cluster singles and doubles with perturbative triples (CCSD(T)) method. Comparison with results obtained with the OEP MBPT(2) exchangecorrelation potentials and some density functionals has shown a qualitatively incorrect shape for some widely used exchangecorrelation potentials. Higherorder response properties, such as hyperpolarizabilities, are described very poorly with conventional functionals. The typical relative errors described in the literature are about 100 percent. The reason for the poor description is an incorrect longrange asymptotic behavior and an incomplete cancellation of the coulombic selfinteraction. The OEP method is free from those drawbacks and calculated values of static hyperpolarizabilities with the exchangeonly potential are close to those derived from HartreeFock theory. The second kernel required for calculations of hyperpolarizabilities within the DFT framework has been derived for OEP potentials by using the developed diagrammatic technique. The structure and properties of the second OEP kernel are discussed. ( en )
 General Note:
 In the series University of Florida Digital Collections.
 General Note:
 Includes vita.
 Bibliography:
 Includes bibliographical references.
 Source of Description:
 Description based on online resource; title from PDF title page.
 Source of Description:
 This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
 Thesis:
 Thesis (Ph.D.)University of Florida, 2007.
 Local:
 Adviser: Bartlett, Rodney J.
 Statement of Responsibility:
 by Denis Bokhan.
Record Information
 Source Institution:
 UFRGP
 Rights Management:
 Copyright Bokhan, Denis. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 670421880 ( OCLC )
 Classification:
 LD1780 2007 ( lcc )

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Full Text 
Equations (155) and (156) are the basis of linear response theory. Since equation (1
55) is not linear with respect to pi the solution should be obtained by some iterative
procedure.
However, for all practical purposes a direct use of equation (155) is not convenient.
To get a more convenient form, the following matrix element should be considered
< plpi(w)l\pi >=< plvl(wi)lpi > + dr~cp(r)( I dr' r) >c~(r) +
+ J dr r)(J dr/'f,,(r, r', U)pIWr', U)) r) (159)
On the other hand, < o, lpi(r, w)  > can be expressed in terms of a response function
OCC unoCC
P,(r, ) = Pa( api + ai(W)ppi (160)
i a
where
Pia() =, i( ( =7 (1 61)
L + (Ea ~ i) (a C i)
Substituting equation (161) into equation (159) we have
OCC unoCC
Pia(W) ( + Fa i <) =< aVI(W) \i > + ([< ab > +
j b
< f f (w) PaPb b + [< OiObjOa >j >+ < I fxc(w) >]Pbj)
OCC unoCC
Pai(W)(Fa )i +i b ([< OiPbOa j >+
j b
< I  fGc(w) \PaPj >]Pjb + [< Pab iPj > < fc(') >]Pbj) (162)
Introducing the notations
Aai,bj = 6ij6ab(Ea Ei)+ < 0aPb > + < f ;.fc(,) PalPb >
Bai,bj < K 1ipObjpaOj > + < f( cl) (cPapj > (163)
When functional derivatives are taken from the diagram (VC4), diagrams (VC41)(VC4
14) (323) will appear.
2 i = b2
ajO
k d
c a
lc iJ c a c
(VC41) (VC42) (VC43) (VC44) (VC45) (VC46) (VC47)
_Ljd dTTT dF^T a ^
S 48 a a a5
=C 1 1C49 CC
(VC48) (VC49) (VC410) (VC411)
(VC412) (VC413)
Diagram (VC5) produces diagrams (VC51)(VC516) (3 24) after differentiation.
c c d
1 d c c ib 7ib c b
bj c b 1 ab a
j b 2i b 2 b k
k (k k 2
(VC51) (VC52) (VC53) (VC54) (VC55) (VC56) (VC57) (VC58)
S c 2 C 1 2 d 2
S bj b jk ab jkab ja b a b
(V 59) (VC510) (VC5) (VC512) (VC513) (VC514)
(VC59) (VC510) (VC511) (VC512) (VC513) (VC514)
(VC515)
(VC516) (324)
Differentiation of diagram (VC6) generate diagrams (VC61)(VC616) (325),
a b 2c bca ba ba ( b
(VC610) (VC61 1) (VC612) (VC613) (VC614)
bi\\ 2 k
aic b b
(VC615) (VC616) (3 25)
k b
1;i
a Fb
(VC414)
(323)
a Mbh
2
(VC69)
AB INITIO DENSITY FUNCTIONAL THEORY FOR OPEN SHELL SYSTEMS,
EXCITED STATES AND RESPONSE PROPERTIES
By
DENIS BOKHAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2007
construction of approximate functionals is the Local Density Approximation (LDA), where
the energy is given through an integral of a local functional of the density. The second
level is Generalized Gradient Approximation (GGA) functionals, which improve upon
LDA by the addition of the dependence on the gradient of the density. The third level,
known as metaGGA, contains also a dependence upon the gradients of the KohnSham
orbitals. For the GGA and the metaGGA there is the possibility to choose a form of a
functional. A number of different forms have been ir.. 1. I A standard way of choosing
the form of GGA and metaGGA functionals is to satisfy a set of theorems, which are
known to be satisfied by the exact functional. Some functionals contain a set of parame
ters to reproduce experimental data. Such functionals are called semiempirical. There are
also functionals which do not contain parameters but satisfy an extended set of theorems
(nonempirical functionals).
KohnSham DFT with conventional functionals can produce very accurate results for
some properties. Values of the total energy, obtained with GGA are usually comparable
with some ab initio correlated results, like MBPT(2) or CCSD. This is a great advantage
of conventional DFT. However, the accuracy of GGA functionals is restricted by its
analytical form. Generalized gradient approximation exchange functionals cannot provide
the exact elimination of coulombic selfinteraction of the Hartree energy. Since the
exchange energy is much larger than the correlation energy, an incomplete elimination of
selfinteraction can significantly reduce the accuracy of the total energy. Particularly, the
impossibility to describe weaklyinteracting systems, bounded mostly by dispersion forces,
arises from an incomplete elimination of selfinteraction and the wrong behavior of the
exchangecorrelation functionals with the separation of the system onto several fragments.
Although the exchangecorrelation energies are quite accurate with GGA functionals,
the corresponding potentials are not nearly accurate, especially in the intershell and
.Ii~,nIl ''I ic regions. As a consequence, the accuracy of the density is much lower than the
accuracy for the energy. Furthermore, the qualitatively incorrect potentials reduce the
Tr6) (r6)
S8(r7 )5 (r7 ) J(
(FX7) 7(FX8) (618)
Functional derivatives from the set (V1)(V6) can be presented by two types of diagrams.
The first type contains the exchange kernel, diagrams (F13)(F18) of (619)
(r6) 1(r7) 6(r6) (r7) a6(r6) (r6)
S(r7 ) (r6) 1 (r7) ( ) f(r) f(r3)
f(r) (r) (f(r3) b fx(r3) ) b (r b) (r,)
fxrr)x3rx) xr7
(F13) (F14) (F15) (F16) (F17) (F18) (619)
The second type contains the exchange potential, giving diagrams (VX1)(VX8) of (620)
/ r4\ r 6 r ) 6r4 r5\ 6 6r4\ r5\ r6) (r\ rT \ 6 )
(r4\ r5\ r6 )a (r4\ r5\ r6 ) (r4\ r \5 r6) 6(r4\5 \ r5 6)
(r4 r5 r6 4 r5 r6 6r5(r4 r \ 6 ) 64r\ r )r6
V V V V
(VX1) (VX2) (VX3) (VX4)
/ 6(r rs rr ) ( r4 r ) 6(r4\ r r6) (r4\ r\ r6 )
Vx ax x x
6(r4\ r5\ r6) a6r\ ( 5 r6) lrrr5\ r6 6 ) ar 64r 54 r 6 )
6(r4\ rs*5 x J ) b'4 s r54\ 17\ r6 ) V ^(4 rs\ r6 )
4 6456 456
(VX5) (VX6) (VX7) (VX8) (620)
The notation r4\r5\r6 means that for each particular diagram all six permutations of r4, r5
and r6 should be taken. The functional derivatives from diagrams (FX1)(FX8) lead to the
set (SX1)(SX14) of (621) and (622).
8/ 4 \ r \ r6) 1 (r4\ 5 6) \A(r4\ r\ r6 ) 6 (r \ r )
S5 6 4 5 6 ) 5r4 6 ( \7 (r5\ r \ r6
S8(r 54\ r5\ r6 4 5 6 )r (\ r55\ r6
k c a b 61r4 175 r
8(r4\ r5 6 r ) 8(r4\ r5\ r6 ra
(SX1) (SX2) (SX3) (SX4) (SX5)
A^ ^ ^ ) ar4 6\ r5\ r6
(r4\ r5\ r6 (r\ r5\ r6
k(r\r5\ r6 )\(r\ r5\ r6) 6(r,\ r5\ r6 ) ( rs r 6
6 )
S (r4 5 r 6 a 6a b k
b k 64) 175 6 k4 (5 ) 4 5 6(
(SX6) (SX7) (SX8) (SX9) (621)
After taking the functional derivative of diagram (VC14), we obtain diagrams (VC141)
(VC1415) (333)
F F
b i Fb 6Z a b a
b a j a ^.
(VC141) (VC142) (VC143)
(F F(F
a ba b k b
(VC147) (VC148) (VC149)
 82 b a i j b
(VC144) (VC145)
a iai
C1410) (VC14
(VC1410) (VC1411)
(VC1413) (VC1414) (VC1415)
The last set of diagrams (VC151)(VC1515) can be produced by
diagram (VC15) (334)
162 (1 62" kF F 53 C F k & F
(F k F F t k
1 1 a a & 1
1i a 1 ai %a61 6, a
(VC151) (VC152) ( 153) (VC154) (VC1
(VC151) (VC152) (VC153) (VC154) (VC155) (
(VC157)
b k
(VC157)
b
(VC1511)
F 1
b 'k c
(VC158) (VC159)
2 6 61
C
differentiation of
P a ,
M'b
VC1510)
VC1516)
F)8 b, F) bk UF k
b( (VC1514)Li (C515 b
(VC1512) (VC1513) (VC1514) (VC1515)
(VC1512) (VC15 13) (VC15 14) (VC15 15)
Together with diagrams (V1)(V6), all diagrams presented in this section form a set, which
is necessary to construct the correlation kernel. The interpretation of all diagrams is given
in Appendix A.
(VC146)
(VC1412)
(333)
(334)
 OEPsemi
UHF
...... MBPT(2)
 CCSD
 CCSD(T)
0.5 1.0 1.5 2.0
R,A
Figure 44. OH potential energy curve.
1'
1I
1'
74.6
74.7 
74.8
74.9
75.0
75.1
75.2
75.3
75.4
75.5
75.6
75.7
7C C
where Nom' is the numerator, with interchanged indexes k and 1.These expressions can be
represented by the following two diagrams (246)
1 1
2
k
k
m* m
<+ <(246)
Namely these two diagrams will appear after taking functional derivatives using dia
grammatic rules for taking functional derivatives. Proof for the case where functional
derivatives are taken with respect to unoccupied orbitals is the complete analog of this
one. Hence, we can formulate a second statement.
Statement 2: When functional derivatives from some vertex of a diagram are
taken with respect to occupied (unoccupied) orbitals, and a line which corresponds to
occupied (unoccupied) orbital inserted, together with diagrams which arise from taking
functional derivatives from denominators; we ahv have diagrammatic expressions, which
corresponds to part of EY )((k)+( Ck))s.
This statement is a direct corollary of the above proved statement about "bracket
type" denominators and statement 1. Since we have correspondence in the second and
nth order,using the method of mathematical induction, it is possible to prove, that the
correspondence taks place in all orders (finite or infinite).
2.3.4 Interconnection in Infinite Order
It should be noted that summation of perturbation corrections to the energy and the
wave functions up to infinite order provide correspondence with the full CI:
00
E n E) E(FCI)
n0
n O (n) (FCI) (247)
n=0
first member of the righthand side corresponds to 60(3), while the second corresponds to
56(1)+Q (2).
To finally establish the interconnection between the two approaches, we need to
show that it is possible to transform diagrams which contains "brackettype denominators
to standard diagrams. In a previous subsection this procedure was described for the
case where one horizontal line crosses the contour in the diagram, which corresponds
to the energy in second order MBPT. We need to prove, that energy diagrams, which
contain contours crossed by m lines, after taking the functional derivative which respect to
unoccupied (occupied) orbitals and inserting lines of unoccupied (occupied) orbitals; can
be transformed to linear combination of m diagrams which do not contain "brackettype"
denominators.
Consider the following diagram (241):
< \ 1
2
k
I m
(241)
Numbers on lines show the conditional number of lines which are entered for simplicity of
further manipulations. The desirable sum of diagrams has the form (242)
2 2 1
k k
m m / / m
S+ + ......... + L k (242)
Together with diagrams which correspond to functional derivatives from denominators,
this set of diagrams forms products like Q(k)+S6(1), where k+l equals the order of MBPT.
In the same way diagram (VC12) generates diagrams (VC121)(VC1216) of set (331)
ij
a
F
b i
F
(VC123)
2 j
2 6
b i
F
(VC124)
S b
F
(VC125)
1 b
F
c
(VC126)
F 1 F
(VC127) (VC128)
aj 6
/b
(VC129)
(VC129)
U ij2
b
i a a
l J
,J a 8 CC
(VC1210) (VC1211) (VC1212) (VC1213)
6 b b
1 Ob
Fb Ca (V7a
(VC1214) (VC1215) (VC1216)
Differentiation of diagram (VC13) gives us diagrams (VC131)(VC1315) (332)
1 2 1 2
b k b k j c j c
82 82 8
j j b b
1 1 a a
a a i a 13
F F F F
(VC131) (VC132) (VC133) (VC134)
b l
2 k
a F
j 6
2 C b
a
F
(Cb 
a F i F
(VC135) (VC136)
0 V1 0i 2 2
b j k i C
b F F
c k
i 82 a 62
a b b
F F 61
(VC137) (VC138) (VC139) (VC1310) (VC1311) (VC1312)
2b1 a 
S61Z 62 b J
b a i b
a(VC1313) (Vc k a(VC1315)
082 Nk c
(VC1313) (VC1314) (VC1315)
(331)
T,
j 6 2 2
(332)
LIST OF FIGURES
Figure page
41 Exchange and correlation potentials of Li atom (radial part). A) Exchange po
tential. B) Correlation potential ............... ... 73
42 Exchange and correlation potentials of 02 molecule across the molecular axis.
A) Exchange potential. B) Correlation potential .... . ... 73
43 LiH potential energy curve. ............... ......... 77
44 OH potential energy curve ............... ......... .. 78
45 HF potential energy curve. ............... .......... 79
51 Exchange potentials of Ne atom, obtained in different basis sets . ... 88
52 A 1II chargetransfered excited state of He ... Be ... . .... 96
53 LUMOHOMO orbital energy difference ................ . 96
[64] S. J. A. van Gisbergen, P. R. T. Schipper, O. V. Gritsenko, E. J. Baerends, B.
C'!l 1,p1. ;iw., and B. Kirtman, Phys. Rev. Lett. 83 697 (1999).
[65] F. A. Bulat, A. ToroLabbe, B. C'!i ip ,1. i B. Kirtman, W. Yang, J. C'!. in Phys.
123 014319 (2005).
[66] M. Kamiya, H. Sekino, T. Tsuneda and K. Hirao, J. C'!. in Phys 122 234111 (2005).
[67] R. J. Bartlett, D. M. Silver, Phys. Rev. A 10 1927 (1974).
[68] H. P. Kelly, Adv. C'., in Phys 12 314 (1963).
[69] R. J. Bartlett, H. Sekino, ACS Symposium Series 628: Nonlinear Optical Materials
23 57 (1996).
[70] A. M. Lee and S. M. Colwell, J. Chem. Phys. 101 9704 (1994).
[71] S. P. Karna and M. Dupuis, J. Comp. ('!I, ii 12 487 (1991).
[72] H. Sekino and R. J. Bartlett, J. ('!,. ii Phys 85 976 (1986).
[73] H. Sekino and R. J. Bartlett, J. ('!,. ii Phys 98 3022 (1993).
[74] D. Bokhan, R. J. Bartlett, J. ('C!. ii Phys. (submitted).
The condition (625) might be used as a hint for developing new density functionals, as
none of the existing functionals exhibit such behavior.
6.2 Numerical Results
As a numerical test the static hyperpolarizabilities of LiH, H20, CO and CN are
calculated. In all cases uncontracted ROOSADZP basis sets[27] are used and equlibrium
geometries are taken from ref[36]. The results calculated with different methods are
presented in Table 61. In the first four columns are static hyperpolarizabilities obtained
Table 61. Hyperpolarizabilities of several molecules (in a. u.)
HF OEPHF OEPx LDA CCSD
LiH
3 312.130 314.910 312.126 621.093 691.406
f3. 201.150 207.868 201.641 404.297 204.171
CO
,,, 31.016 24.864 29.455 66.406 27.343
0.. 3.073 4.660 3.118 13.203 5.859
CN
0.. 1.299 0.211 3.906 1.953 164.02
A,, 17.651 15.646 18.121 62.500 68.359
H20
03. 6.713 6.239 7.812 13.672 5.859
j3. 0.497 0.604 1.215 6.138 3.870
3yy, 10.866 10.32 11.847 23.437 7.645
with HartreFock nonlocal exchange and with the two local exchange operators: OEPHF,
which means equation (65) without the second kernel term, and OEPx, which means
all terms in equation (65). The hyperpolarizability values calculated with the OEPx
method are in good agreement with the corresponding HartreeFock values, and generally
no better. This good agreement can be explained by the fact that OEPx method is free
from the selfinteraction error, has the correct longrange ., ii1! i .I ic behavior, and the
correct HOMO values. As hyperpolarizabilities are response properties of third order their
calculation within the DFT method requires very precise exchangecorrelation potentials.
Only potentials which simultaneously satisfy all theorems and conditions pertaining
to the DFT exchange potentials can be expected to reproduce even the HartreeFock
Using the effective operator approach it is possible to define VC7) in all possible orders.
This procedure for an infinite sum of orders gives Vxc. Inserting this potential into the
KohnSham equations, we can find p("),which corresponds to the oneparticle density in
the full CI method.The same density could be used in a method like ZhaoMorrisonParr
(ZMP)[19] to extract the corresponding exchangecorrelation potential. Since the full CI
energy does not depend upon the choice of orbital basis set, the OEP procedure[13, 15]
cannot be used directly for this case. The infinite sum of all energy corrections does not
depend upon choice of orbital basis set, but each term of this sum does depend upon
the choice of orbitals. This fact enables us to consider all term of the infinite sum of
energy corrections as orbitaldependent functionals. After the construction of the set
Vx1) .... t) and summation up to infinity, we will have the exchangecorrelation potential
which corresponds to the full CI. Using the equivalence of the functional derivative and
the effective operator approach, it is possible to conclude that after summation of this
set of potentials, we again will have the same result as in the effective operator approach.
Redefined in such a way, the OEP procedure[13, 15] for the full CI energy produces the
same density as the ZMP method[19] would from full CI.
APPENDIX B
INTERPRETATION OF DIAGRAMS OF EXACTEXCHANGE SECOND KERNEL
For the calculation of last term in equation (65)an auxiliary basis set decomposition
is used
< 'Oaa'(ha Oc Iga r OkO >= < OaOiXA >< 1 TiXp >< 'Pco'CkuXv > g\Pv (B1)
where X, are the auxiliary functions. The expression for g\,, can be written in the form
(B2)
gA/\ = Z(X )A(X ),(X )"r + Z(X1)A(X1), +
Ki7,rl K,r
(X 1)(X1),)A( + (X 1)A(X ),,o (B 2)
where
X, = 2 < ajA>< a (B3)
Ei Fa
Two separate g\, should be constructed for each spin component. Expressions for 0,,,,
OXA, and 0,,, can be written in terms of the diagrams (Fl) (F18).
,,, 2_ f ia,(< 'PaPjK >< ', .'rlI > + < Pa pjrI >< ', .'.. >)
ija Ea)(Ej Ea)
2 fiav(< ibbK >< Pa brl > + < bPil >< aIbK >)
2
a(Eb Ea)(Ei Eb)
i,a,b
+2C fijv < OaOiK >< Oa)jrl > 2 fabv < 2i(4bK >< )Pia > (B4)
( a)( a) ab ( Ea)(E Eb)
where the integrals involving the exchange kernel can be calculated using the formula
fpqv >= < q > (X 1)AQpX 1)P/ (B5)
KAp
When rules for taking functional derivatives diagrammatically are applied to this set of
diagrams, we will have the next set of diagrams (237)
2 + 2 +2 F + 2 ((237)
The first two diagrams are equal to diagram 16; the third and fourth equal to diagrams
17 and 16 respectively. Since the lefthand site of eq.12 can be represented by diagram
3, we have an exact equivalence between the functional derivative and the effective
operator approaches in the first and second order for the "Kohn'lI ii partitioning of the
hamiltonian.
2.3.3 Interconnection in Higher Orders
Equations for the effectiveoperator approach in the nth order of MBPT have the
form (238)
< (KS (1)+.... (n) (+)Ap(r)((1) .... (n))IK > 6(6Q(k)+Q(nk)) = 0 (238)
k0
This equation tells as that we need to collect and equate to zero the sum of all possible
combinations of wave operators and density correction operators, and these combinations
must be of nth order. The same result should correspond to taking functional derivatives
from all energy diagrams of nth order.
Statement 1 : When functional derivatives from some vertex of a diagram are
taken with respect to an occupied (unoccupied) orbital, and a line, which corresponds to
an unoccupied (occupied) orbital inserted, we .i. ,i will have diagrammatic expressions
which correspond to part of k 0(Q(k)+2Q(nk))sc.
Proof: When functional derivatives are taken with respect to an occupied (un
occupied) orbital, and a line which corresponds to an unoccupied (occupied) orbital is
inserted, we alv,v have a contraction of the delta function with some fully connected
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Yarkony (World Scientific Publishing Co., Singapore, 1995).
[3] F. E. Harris, H. J. Monkhorst, and D. L. Freeman, Algebraic and Diagrammatic
Methods in M ri,;Fermion Theory (Oxford University Press, New York, 1992).
[4] P.O. L6wdin, J. C'!. in Phys. 19, 1936 (1951).
[5] P. Hohenberg and W. Kohn, Phys. Rev. 136 B864 (1964).
[6] W. Kohn and L. J. Sham, Phys. Rev. A 140 A1133 (1965).
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[8] E. K. U. Gross and W. Kohn, Adv. Quantum. ('C!. ii, 21,255 (1990).
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D. P. C('!..i(World Scientific, Singapore, 1995).
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(1996).
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Inc, New Jersey, 2004).
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(2002).
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[16] R. J. Bartlett, V. Lotrich and I. Schweigert, J. C'!. iin Phys 123, 062205 (2005).
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(2002).
Using equation (213) it is possible to write down the functional derivatives from some
denominator in the general form:
De K1 K1 Ka Ka
4 (_  Ii2 12 2 )1jl 2 c" 12 > ia Y ja +.+
V, / I ( occ I uno I J ( occ unocc) ..... +
i=1 j=1 a,a#1 i=1 j=1
K/ K3 Ka Ka
... ...+ ( 2 _ E I2) E E (228)
i=l j=1 a,a43p i=l j=1
Now it is possible to write down a term which includes the functional derivative from the
denominator:
Norm Den
(Den)2 6V,
KT(vi3 1<3 2 KB3 2
Socc j noocc j u(2n29)
We now have the possibility to formulate how to take functional derivatives from denom
inators. To take functional derivatives from denominators, all diagrams, where one of the
lines is doubled and between these lines which arise from doubling, the corresponding
diagonal 6function is inserted. This procedure must be provided for all horizontal lines on
the diagram and for all the contours the lines cross.
2.3.2 Diagrammatic Functional Derivatives in SecondOrder ManyBody
Perturbation Theory
Direct interconnection in first order is clear from diagram's 2 and 3.The lefthand
side of equation (217) is equal to diagram 2, the righthand site is equal to diagram 3
when the functional derivative has been taken from the expression for the exchange energy.
The diagrammatic expression for the secondorder energy in the general case has the
form[l]:
F
E(i b a b (
E(2) F +2 + 2 (230)
BIOGRAPHICAL SKETCH ................... .......... 135
. i ,! II dll ic behavior, achieve by the ColleN. bet algorithm, makes it possible to avoid an
underestimation of Rydberg state energies and overestimation of the polarizabilities. This
overestimation for polymers, particularly for hyperpolarizabilities, has attracted a great
deal of attention[6466]
The problem of chargetransfer excited states in TDDFT relates to the inability of
the zerothorder orbital energy approximations, as the HF virtual orbitals and TDOEP
excited orbitals show qualitatively different behavior. If the HF and OEP operators were
the same, a proper treatment using the whole space should be able to overcome this
limitation, even if the usual zerothorder approximation offered by TDDFT would be
a poor one. However, this difference arises from the form of the operator, local versus
nonlocal, and, as such, remains a fundamental problem for the DFT method itself, as has
been observed [51]
In conclusion, for all the properties considered in the paper, except for charge
transfer excited states, there seems to be no significant difference between the results of
TDHF and exchangeonly TDOEP, as one might expect. Hence, the focus should remain
on the correlation potential as it is in ab initio dft[16, 17, 48, 62].
Table 31. Orbital energies and zeroorder approximations to excitation energies
Exchangeonly OEPMBPT(2)
orbital IP[28] Orbital a, FHOMO Orbital Fa FHOMO Excitation
energies energies energies(exp)
2s 48.42 46.19 43.25
2p 21.56 22.51 20.23
3s 5.15 17.36 7.58 12.65 16.85
3p 2.59 19.92 3.72 16.51 18.70
4s 1.95 20.56 2.66 17.57 18.73
3d 1.49 21.02 2.05 18.18 18.97
[28] This compares to the OEPx value of 22.51 and OEPMBPT(2) value of 20.23. The
remaining unoccupied, but negative energy 3s orbital is changed by over 2 eV due to the
MBPT(2) correlation. The OEP2(sc) ab into dft value changes this to 5.18 eV, attesting
to the poor convergence of the standard KS partitioning of the Hamiltonian. Once
OEP2(sc) calculations are done for the 3p, 4s, and 3d states, there is similar agreement
between the OEPx and OEP2(sc) results, contrary to those shown in Table 32 for the
standard (ks) choice.
Table 32. Excitation energies of Ne atom using OEPMBPT(2) KohnSham orbital
energies
Term EOMCCSD TDDFTexchangeonly TDDFT OEPMBPT(2) Exp[29].
3P 16.5353 17.1748 15.2132
1p 16.7274 17.5209 11.4434 16.847
3 18.8303 19.8681 16.3320
1 18.8822 19.9181 16.9465 18.703
3P 18.9156 19.9134 17.5645
1p 18.9791 19.9134 17.7686 18.725
S 19.5118 19.8811 17.t. ;
S 19.9323 2', ' ; 17.3454 18.965
3P 20.3747 20.4905 18.3455
1p 20.5588 20.6090 18.8788
Despite the differences, at least both the exchangeonly and the OEPMBPT(2) give
qualitatively correct results for the Rydberg series. Helping to ensure this is the fact that
the exchange potential has the correct .,ii iil '1 i c behavior and the exact cancellation
To make the transition from an infinite basis to a finite computational basis, one
introduces the matrix representation of the OEP equation. Eqn (518) then becomes,
w(Vx + K) 0
(524)
(525)
w = aiR1
Using weighted least squares
6Vx[) :,ia < ilVxa > +Kia22] 0
(526)
gives the OEP equation, Eqn (516).
As is customary in practice, the computational form for the OEP equation can be
obtained by twice applying the oneelectron projector,
0(1) = nr] >< rn\rl >'< r]1
where  r) defines an auxiliary basis, respectively, for electron 1 and 2. Without restriction
the basis can be assumed to be orthonormal The socalled outer projection[53] of the
response function is then
X,, = (ai )(iav) /(ci ea)
(527)
Using this expression and insisting upon the linear independence of I ry) gives the OEP
equations in the form,
< lX(1,2) >< lVx(2) + oK(2))
XVx
Y,
Vx
OVx1
Y
S (pai) (ijla)
S(i ja)
X1Y
where the o indicates the product of X and K in Eqn (518) that results in the Y column
matrix.
(531)
Variation of the energy can be presented in following form:
>uJ=C
dr2 6~p(r2)
6p,(r2)
(27)
We can write down variations of the orbitals and the KohnSham potential too:
(28)
(29)
Using equations (28) and (29) we can rewrite the variation of the energy in the following
form:
6E ") r= dr2
S 4p(T)
dr6p, (r2)
dr6V(r
Vs(ri)
V (rl)
dr 6p(r)
6p(r)
Now we can write down the chainrule[13, 14] for the calculation of Vc) :
6E ")r 6E ")
6Jp(r) 6 6p(r2)
6rp,(r2) 6V,(r3)
6Vs(r3) 6p(r3)
S 6E( 6V(r3)
dj (V() ( p(rl)
Taking into account the fact that[14]:
6Vs(ri) C p ,C
p,q p
6Fr
6v,(r)
W6V(r2)
6V(r2)
it is possible to write an explicit expression for V~) (r) [13]
V(")(r) 2( J
p,q p
Using the fact that
dr26E () dr3 p (r3) q (r3)X (r, r3)
Op,(r2) Jp Eq
(215)
dr2X(r1, r2)X (r2, r3)
(210)
(211)
X(r, r2) 2 z ) i( ) c)a(rl)i(r2)>a(r2)
(212)
(213)
(214)
6p(2f d p2 6V/(r) (
6V,(r) dr 6 p(r) 6p(r)
j 6P(r)
S,(r) ,(r)
6(r1 r3)
(216)
TABLE OF CONTENTS
page
ACKNOW LEDGMENTS ................................. 3
LIST OF TABLES ....................... ............. 7
LIST OF FIGURES .................................... 8
A BSTR A CT . . . . . . . . .. . 9
CHAPTER
1 INTRODUCTION ...................... .......... 12
1.1 Ab Initio WavefunctionBased Methods .................. 13
1.1.1 HartreeFock Method ......... .......... .... 13
1.1.2 ElectronCorrelation Methods ....... ............ 16
1.2 KohnSham Density Functional Theory ............. .. .. 19
1.2.1 TimeDependent Density Functional Theory . . 21
1.2.2 TimeDependent Density Functional Theory Linear Response Theory 23
1.2.3 Problems with Conventional Functionals . . ..... 26
1.2.4 OrbitalDependent Functionals ................ .. .. 28
1.2.5 Ab Initio Density Functional Theory ................ .. 29
2 INTERCONNECTION BETWEEN FUNCTIONAL DERIVATIVE AND EF
FECTIVE OPERATOR APPROACHES TO AB INITIO DENSITY FUNC
TIONAL THEORY ........... ..... ........... 31
2.1 Equations for the ExchangeCorrelation Potential in the Functional Deriva
tive Approach ........ .. ...... ............... .. .. 31
2.2 Equations for the ExchangeCorrelation Potential in an Effective Operator
Approach ................... . . .... 34
2.3 Interconnection in Arbitrary Order ............ ...... 37
2.3.1 Diagrammatic Functional Derivatives . . . .... 37
2.3.2 Diagrammatic Functional Derivatives in SecondOrder ,M ,ivBody
Perturbation Theory .................. ....... .. 38
2.3.3 Interconnection in Higher Orders .................. .. 41
2.3.4 Interconnection in Infinite Order .... . .. 45
3 AB INITIO TIMEDEPENDENT DENSITY FUNCTIONAL THEORY EM
PLOYING SECONDORDER MANYBODY PERTURBATION OPTIMIZED
EFFECTIVE POTENTIAL .................. .......... .. 47
3.1 Diagrammatic Construction of the ExchangeCorrelation Kernels . 48
3.1.1 Formalism ................ .... . . 48
3.1.2 An Example: Diagrammatic Derivation of ExchangeOnly Kernel 52
4.2.3 Dissociation Energies
Dissociation energies, calculated with semicanonical OEP and PBE are presented
in Table 43. Potential curves for the LiH, OH and HF molecules are shown on Fig 43,
44 and 45. For all curves the dissociation energies have approximately the same level of
accuracy as in the MBPT(2) case, but the semicanonical OEPMBPT(2) improves the
shape of the curves.
Table 43. Dissociation energies (in kJ/mol)
PBE OEPsemi Exp.
N2 1013.27 964.56 941.64
02 598.14 549.12 493.59
CO 1122.16 1122.00 1071.80
CN 821.151 794.51 745.01
4.2.4 SingletTriplet Separation in Methylene
Results for the extensively studied splitting of singlet and triplet states of methylene
[41, 42] are reported, using the uncontracted ROOSADZP basis sets: 13s9p3d for carbon
and 8s4pld for hydrogen. Equilibrium geometries for both states are taken from reference
[41].The energies for the two states, calculated with different methods are presented in
Table 44. There are significant differences in the energy separation compared to the
experimental value (8.998 kcal/mol) for all of the presented methods. This is a basis
set issue, but can also be explained by the fact that the singlet state of methylene has
a significant contribution from twodeterminants[41], so MBPT(2) is a much poorer
underlying approximation for their difference than infiniteorder CCSD or CCSD(T), and
especially the twodeterminant CCSD (TDCCSD) results[41]. Where the absolute values
of PBE energies are not too good for the two states of CH2, the difference is consistent
with MBPT(2) and its OEPgenerated ab initio dft form.
4.3 Conclusions
Ab initio dft calculations with OEPMBPT(2) semicanonical potentials show
significantly improved results over OEPMBPT(2) with the KohnSham partitioning of
while diagram (VC7) gives diagrams (VC71)(VC714) (326).
2 i 1a
(VC71) (VC72)
(VC73) (VC74) (VC75) (VC76) (VC77)
(326)
(VC78) (VC79) (VC710) (VC711) (VC712) (VC713) (VC714)
Finally, diagram (VC8) produces diagrams (VC81)(VC814) (327).
(VC81) (VC82) (VC83) (VC84) (VC85) (VC86) (VC87)
(VC88) (VC89) (VC810) (VC811) (VC812) (VC813)
(VC814)
(327)
In the same way the rest of the diagrams should be differentiated. Since the Fock operator
depends upon occupied orbitals, Frings on the diagrams (VC9)(VC15) must also be
differentiated. Diagram (VC9), after taking functional derivatives, produces diagrams
(VC91)(VC910) (328).
a 6 a
2 k2 az2
(VC91) (VC92)
k F 6
F F b
(VC96) (VC97)
(VC96) (VC97)
F(
b 6
F
(VC93)
F
(VC98)
(VC94) (VC95)
b k
22
V 2 a J6
(VC94) (VC9105)
(328)
of an exact mapping between densities and external potentials. In the ground state
formalism, the existence proof relies on the RayleighRitz minimum principle for the
energy. A straightforward extension to the timedependent domain is not possible since
a minimum principle is not available in this case. The existence proof for the onetoone
mapping between timedependent potentials and timedependent densities, was first given
by Runge and Gross[7].
We can start from the timedependent Schr6dinger equation
at H( (t)= (t) (1 42)
evolving from a fixed initial manyparticle state
((to) = o (143)
under the influence of different external potentials v(r,t). For each fixed initial state To,
the formal solution of the Schr6dinger equation (142) defines a map
A: v(r,t) + T(t) (144)
between the external potential and the corresponding timedependent r: ,_viparticle wave
function and a second map
B : (t) p(r, t) =< T (t) &(, t) (t) > (145)
Thus, the following mapping can be established:
G: v(r, t) p(r, t) (146)
The RungeGross theorem establishes that the G mapping is invertible up to some
additive, timedependent constant. In other words, two densities p(r, t) and p'(r, t)
evolving from the common ground state To under the influence of potentials v(r,t) and
However, the number of possible excited determinants grows rapidly with the
number of electrons and basis functions in the system. Therefore, the full CI method
is computationally intractable for any but very small systems. Among the approximate
electroncorrelation approaches, the most common are the Coupled Cluster method[2]
and Many Body Perturbation TI. ii [ :]. Any truncated version of the CI method
has a qualitatively wrong behavior of the energies and wave functions while increasing
the number of particles in the system. Therefore, the CI methods with limited level of
excitations cannot be used for highlycorrelated systems. The Coupled Cluster method
and Many Body Perturbation theories are free from this lack of extensivity failure and
are very common for the molecular computations. In some cases perturbation theory
can provide an accurate description of electroncorrelation effects at a much lower cost
than necessary for the Coupled Cluster method. The secondorder RayleighSchr6dinger
perturbation theory is the simplest and least expensive ab initio method for taking into
account electron correlation effects.
In this perturbation theory the solution of Schr6dinger equation
Hf = E' (116)
can be found using the Slater determinant as a reference. Generally, such a determinant
may be constructed from the orbitals, generated by some oneelectron operator
hop = (1V2 + u)yp = pp (1 17)
2
The first step of any perturbation theory is the partitioning of the Hamiltonian into a
zeroorder Ho and perturbation
H = Ho + V (118)
where
elec
Ho = Eo = (hi)K = ( F)K (119)
i i
this zerothorder difference should be a decent approximation to the excitation energy, and
is, as shown elsewhere [16] when accurate exchangecorrelation potentials are used. In this
sense, the orbital energies in KSDFT should have a certain meaning.
Furthermore, as pointed out in ref [16], we can also consider this equation to offer
a Koopmanslike approximation to the principal ionization potentials, since, barring
pathological behavior, when we allow an electron to be excited into the continuum, its
orbital a will then have no overlap with the bound orbital matrix elements of hKS and
the kernel. Consequently, we are left with nothing but ci in the TDDFT equations. So in
this 'sudden', adiabatic approximation, the KS orbital energies should offer an estimate
for each of the principal Ip's, not just the highestoccupied (homo) one. When based upon
the relatively correct V,, obtained from ab initio dft[16, 17] this estimate is superior to
Koopmans' theorem for the homo and the first few valence Ip's, but is inferior for the core
or'.il '1[21]. See also C('l.: et al [25]. However, the OEP2 semicanonical (sc) ab initio
dft [16, 17] approximation has the distinct advantage that it uses a much better behaved
unperturbed Hamiltonian than the usual KS choice, Ho = : hKs(i).
To illustrate the evaluation of the kernel and the solution of TDDFT equations, we
consider the Ne atom. All the excited states in Ne correspond to Rydberg excited states.
To obtain reasonable values requires a quite extensive, diffuse basis set. We choose to
start with the ROOSATZP atomic natural orbital '. ,[[27] consisting of (14s9p4dlf)
primitive gaussian functions contracted to a [5s4p3dlf] set. This underlying basis was
then augmented by a set of eventempered diffuse functions [3s3p3d], with exponential
parameters = ,' a8 = 0.015, a = 0.013, ad = 0.012, b=1/3. The auxiliary basis is
chosen to be the same, but without the p and d diffuse functions, since for the description
of the exchangecorrelation potential, diffuse functions are not necessary. The orbital
energy estimates are shown in Table 31.
The ionization potential equation of motion coupledcluster (IPEOM CC) result
in this basis for the Ne homo Ip is 21.3 eV with the experimental value being 21.5645
where
Ro (130)
Eo Ho
and Q 1 I >< K is the projector onto the complementary space of P >. Since Ho is
diagonal in the basis of Slater determinants, it is possible to write
R0 > ti i <1 j y j (1 3 1 )
Ro = Eo E, (31)
n 0
For the special case of the HartreeFock reference determinant
Eo + E(1) =< liHo01 > + < 1V1 >= EHF (132)
the secondorder correction to the energy is
10CC uOCC .
E (2) 1 oc < zjlab > (< ijIab > < ijlba >)
2 E EY E. Eb
i,j a,b
For the case of a more general reference determinant, constructed from orbitals of Equa
tion (117), the secondorder correction to the energy has the following structure
ocC unoc 12 occ unocc .
E(2) I < i h fla > 1 1 < ijlab > (< ijab > < i(ba> 34)
i a i,j a,b
1.2 KohnSham Density Functional Theory
Density Functional Theory is an alternative approach to the description of the
electronic structure of molecular and solidstate systems. This method uses electronic
density instead of the wave function as the basic object of theory. The mathematical basis
of DFT is provided by two theorems, introduced by Kohn, Hohenberg and Sham. The
first one, known as HohenbergKohn theorem[5], establishes a onetoone mapping between
the groundstate electronic density and the external potential. The external potential
defines a particular object (atom, molecule, etc) and, because of the onetoone mapping,
the density contains all the information about the system. In particular, the groundstate
energy can be written as a functional of the density. To get the ground state energy in
Finally, there is no doubt that the standard equation of motion coupledcluster
methods (EOMCC)[31]) are less timeconsuming than is the present calculation. As long
as the kernel for a rigorous orbitaldependent correlation potential is this complicated,
twoparticle wave function theories like EOMCC are both superior and easier to do. But
before we can further exploit the interface between DFT and wave function theory to the
benefit of both, it is requisite to have the orbitaldependent kernel defined. This, and its
initial evaluation, is what this chapter offers.
6 2 (< calV.cb > +Kab) F5
i,j,a,b ( E )(E b)(j a)
F5 (< pacjK >< CcjT >< > < ai > + >< a >< i > > +
< Pa ( >< i pj T1 >< ib> + < (ij >< K, ,C > +
< Pja(~C >< (i~(jK >< ibi ab)l > + < (~a(jri >< (jCi(O >< ()(bK >) (B11)
D7 2 (< Va V b > +Kab) F6
(,jki Ei EbEi E
< ><(b c i)(j b)( a)
F6 7 (< kpip,K >< jpbcl >< Xpipa,( > + < icK: >< bc, >< cica > +
< Pi Pc >< b (Pcr >< i~aK > + < b 7l >< b()O{cK >< r r ,C > +
< pi >< bO >< Ci Cal > + < bOcX >< bO >< ccaK >) (B12)
( < oi o > +Kij) F2
D8 = 2 (B13)
i Ek Ea) E Ea) (Ek Ea)
2,j,k,a
The second part is represented by the sum of diagrams (SX1) (SX6)
D92 < ^Oia^fcO > *F7
(,Kjkab K (Ek Eb )(Ej b)(E Ea)
F7 = (< Ob i.' >< OkTl >< a( > + < Wb i. >< kcOck >< aiOrT > +
< P0bPk( >< (c(jTl >< iaK > + < (j(c >< (~Cc( >< Xia( > +
SD10= 2 < Pi Pa PjPb > F8
ijab,c Ec( Eb)(Ei Ea)
F8 = (< ;. / >< cWbl >< Oa( > + < (r,', I r i'< c X Wal > +
< OPjc >< OcC cl >< Oi OaK > + < i jscrl >< PcObK >< Oa( > +
< (PjPcc >< OPcObK >< Oal > + < ijOPcjl >< a rl >< Oi aK >) (B15)
exchange potentials (Fig 41 and 42) is guaranteed by the use of the ColleN. 1 et seed
potential algorithm[30]. In all our calculations a Slater potential[38] was used as a seed
potential, since the ColleN. 1 i algorithm requires the seed potential to be as close as
possible to the actual potential. Two separate Slater potentials, one for alpha and the
other for beta, were used. For all the closedshell systems from Table 41, the numerical
 OEPsemi alpha
 OEPsemi beta
 PBE alpha
 PBE beta
1/r
2 3
R,a. u
Figure 41. Exchange and correlation potentials of Li atom (radial part). A) Exchange
potential. B) Correlation potential
R, a. u.
1 
2
3
4
R, a. u.
6 4 2 0 2 4
R, a. u.
Figure 42. Exchange and correlation potentials of 02 molecule across the molecular axis.
A) Exchange potential. B) Correlation potential
results obtained from OEPMBPT(2) with the semicanonical potential and from DFT
(VC8 13) + (VC8 14)
< ablik >< ijlba >< ckA >< cjp >
(Ei + k Ea Eb)(Ei + sj a b)(Ej 
(VC9
2 f faifkj(< ijA >< akp >
i,j,k,a ( a)A (
faifbj(< ijA
i,j,a,b
(VC9
>< abpi >
 Ea) (Ej 
) (A70)
c)
1) + (VC11 8)
+ < akA >< ijp >)
,a)(Ek a)
2) + (VC10 3)
+ < abA >< ijp >)
a) (Ej Eb)
(A71)
(A72)
(VC9
ijA >< abp >
(i F)(i
3) + (VC10
+< abA ><
E Eb)
2)
ij >) (A73)
(A73)
(VC9 4) + (VC12 5)
2 Y faifba(< ijA >< bjp > + < bjA >< ijp >)
i,j,a,b (i Ea)(E E(E Eb)
(VC9 5) + (VC9
fai0fka(< ijA >< kjp > + < kjA ><
,j,,a (E Fa)(j Fa)(Ek a)
. j, Ak, a
6)
ijp >)
(VC9 7)
Sfaifaj < jbp >
( ija) ( a) ( b
i,j,a,b
(VC9 8) + (VC15 3)=
4 fai < jblak > (< ijA >< kbp > + < kbA >< ijp >)
,j,k,a,b ( Ea)(E Ea)(k b)
2
a,b,c,i,j,k
f, fbj(<
i,j,a,b
(A74)
(A75)
(A76)
(A77)
we can rewrite equation (215) as:
Y o.a(ri)o(ri) < t 1Vnloa > > < E(>) I p(F3)tq(3) (2 17)
a i Fpa q p p 
a,i p,qlP
Using the fact that orbitals p and q can be occupied or unoccupied,we can separate the
righthand site of equation (217) into four parts:
Pa(ri)(Pt() < iK(V a > E(n) 3)(> (r3)
a Fai 6o( Fi  F
a t a Fi 6 o Fi Ft
a,i J 4,i
6E(") ((3)Yb(r3) (2
+ > < I> a b + c.c (218)
abea (pa Fa Fb
a,b740
This form of equation ( 218) will be the basis for the formulation of the diagrammatic
rules for taking functional derivatives and for making the connection with an effective
operator approach to the OEPMBPT exchange correlation potential.
2.2 Equations for the ExchangeCorrelation Potential in an Effective
Operator Approach
Let us consider a oneparticle density operator in secondquantized form:
p(r) = < v(r rl)Pq > a+aq (219)
p,q
Using Wick theorem,we rewrite this operator in normal form with the KohnSham
determinant as the Fermivacuum:
p(r) = < Pp J(r rl)Jl q > {a+aq}+ < LKs 6(r rl)\l KS >
p,q
Nocc
< Kp 1 \(r ri)Pq>< > IP)2+ a a(I)+ (220)
p,q i=1
The first member of the previous expression will be called the /. ,, .:/ correction:
Ap(r) = < ppJ6(r rl), > {aqa,} (221)
p,q
6.3 Conclusions
We have presented the analytical expressions and implemented them for the calcula
tions of static hyperpolarizabilities applicable to general molecular systems with the exact
exchange functional of KohnSham DFT through the TDOEPx formalism, the first step in
the application of ab initio dft[16] It has been shown that the rigorous exchange treatment
in TDOEP and TDHF provide hyperpolarizabilities that closely agree with each other, as
one would expect, although this will not happen for chargetransfer excited states[51, 74].
This clear agreement with TDHF is in contrast to conventional exchange functionals
that severely overestimate hyperpolarizabilities. OEP potentials and kernels are not only
capable of describing excitation energies, but also higherorder properties. Since none of
the standard DFT functionals shows the correct analytic properties of the second kernel,
the OEP formulas and results can be used for the testing and calibration of new density
functionals.
exchangecorrelation potentials are produced by the functional derivative and density
condition approaches.
The timedependent extension of OEP in the adiabatic approximation with
exchangeonly potentials and kernels was recently implemented and applied to some
molecular systems. The corresponding excitation energies and polarizabilities are in good
agreement with timedependent HartreeFock results. However, such results cannot be
used for comparison with experimental values due to lack of a description of electron
correlation effects. In order to account for correlation effects, a correlation kernel corre
sponding to the MBPT(2) potentials has been derived. Its structure and properties are
described in detail. The first numerical results for excitation energies with pure ab initio
potentials and kernels are presented.
The recently implemented OEP MBPT(2) method has been extended to the case
of spinpolarized openshell systems. The total energies obtained for several openshell
systems are very close to the corresponding values obtained with the highlycorrelated
coupled cluster singles and doubles with perturbative triples (CCSD(T)) method. Com
parison with results obtained with the OEP MBPT(2) exchangecorrelation potentials and
some density functionals has shown a qualitatively incorrect shape for some widely used
exchangecorrelation potentials.
Higherorder response properties, such as hyperpolarizabilities, are described
very poorly with conventional functionals. The typical relative errors described in the
literature are about 100 percent. The reason for the poor description is an incorrect
longrange .,imptotic behavior and an incomplete cancellation of the coulombic self
interaction. The OEP method is free from those drawbacks and calculated values of static
hyperpolarizabilities with the exchangeonly potential are close to those derived from
HartreeFock theory. The second kernel required for calculations of hyperpolarizabilities
within the DFT framework has been derived for OEP potentials by using the developed
where
elec Nucl
>I: Iri RAI
i A
VH(r) yC f *(Fi)4)(ri) dr p(ri) dr
) rll /r i ^1
elec
p(r) = (T) i (r) (110)
The HartreeFock exchange operator nix is nonlocal, i. e. it cannot be presented as an
analytical function of spatial variables. However, it is possible to write its action on some
orbital pi
vnlxi(r) ji(/ )P1 (r)dri (1 11)
Since the Fock operator f is hermitian and vH(r) and vnixi(r) are invariant with respect
to unitary transformations of orbitals, Equation (18) can be rewritten in the canonical
form
fPi = Eii (112)
This form of the HartreeFock method is most common for practical implementations.
The HartreeFock method is a system of integrodifferential equations, which cannot
be solved analytically. Iterative methods can be used for the approximate solution of
the HartreeFock equations with a given accuracy. As a first step some guess of orbitals
should be assumed and substituted into the equations. During the second step the
system of equations is solved and a new guess for the orbitals is obtained. Then those
new orbitals should be substituted into the equations again, until the total energy and
density at successive iterations differs less than a required accuracy. This is known as the
selfconsistent field (SCF) method.
For practical purposes molecular orbitals are usually presented as being decomposed
into Gaussian functions, centered on the nuclei. In this case the number of solutions is
much larger than the number of electrons. To decide which orbitals should be included
the X based OEP have no problem even for this situation.
As seen in Table 52, even the notoriously unstable secondorder OEP procedure,
based on HOK has no particular difficulties, either. Since the direct minimization proce
dure of Yang and Wu[46] is based on the same equation as the regular OEP method, all
of the problems, described in ref [49], are caused by using the BFGS algorithm instead
of solving the OEP Fredholm equation with the SVD procedure for the X. That might
appear to be surprising since Ch = l, would mean that the resolvent operator is manifestly
singular, clearly not that appropriate to a nondegenerate perturbation theory functional,
which has alv,i been the intent of such orbital dependent expressions. However, it is also
apparent from the standard, chainrule differentiation used by Rohr et al[49], namely
Vxc(1) Exc/6p(1) = d2d3[6Exc/5,p(2)][6p(2)/6v,(3)][6vs(3)/6p(1)] (532)
that the derivative would not be defined from the beginning as 5Exc/56p(2) has to be
singular when cHOMO = LUMO. For comparison, the OEP2sc result from ab initio dft is
also shown, compared to the normally, poorly converged OEP2 based on the HKS
Other simplifications in the OEP solution are potentially possible. The most obvious
is to make an Unsold, constant energy denominator approximation(CEDA), replacing i 
a C v,9.(This should be contrasted with that of KreigerLiIafrate (KLI)[44] who made
such an approximation in the Green's function where the denominator consists of Cp Cq
before simplifying the from to the occupiedvirtual separation invoked here[16, 54, 57].)
See Gritsenko, et al [56]. Such a constant energy denominator is a relatively painless
approximation here, since Cav appears on both sides of eqn. (18), making it disappear
from the equation for Vx. Once that is done, a resolution of the identity can be invoked to
eliminate the virtual orbitals to give,
(ip la)(aK + Vxi) ({(ip(K + Vx)i (iplj)(K + Vxi) (533)
i,a i ij
(VC14 13) + (VC15 15)
v4 ^ < cjlka >< ablij > (< ibA >< cky > + < ckA >< ibp >)
*4 ,(A118)
bcijk (E Eb)(EF Ea) (Ek Ec)
(VC14 15)
2 < ablji >< kilac >< jbA >< ck > (A 9)
ab,c,i,j,k (i a)( Eb)( c)
(VC15 7) (VC15 8)
8 fa < ablij > (< bkA >< jkp > + < jkA >< bkp >) (A 20)
i,j,k,a,b ( i Ea)(Ei Eb)(E k Eb)
(VC15 9) + (VC15 10) =
8 fai < ablij > (< cjA >< bcp > + < bcA >< cjp >) (A 2
aL,(Ei E,)(E Ec)(Ej Eb)
(VC15 11)
8 Y fai < aklij >< jbA >< kb >(A 22)
8^ (A122)
ijkab (Ei Ea)(Ej Eb)(Ek Eb)
(VC15 12)
8 z fa < ablic >< bjA >< cj > (A 2)
abci,j ( a)( Eb)(EF Ec)
(VC15 13)
16 < ablij >< cika >< bjA >< ck > (A 24)
a,(Eib,c,i,j a)( b)(Ek c)
(V1) + (V2)
2 < ilVcla > (< abA >< i > + < bA >< ab >) (A 25)
,b ( a)( b)(A125)
i,a,b
ACKNOWLEDGMENTS
I would like to thank professors Henk Monkhorst and So Hirata for helpful discus
sions. I want to thank also Dr. Igor Schweigert and Dr. Norbert Flocke for the helping me
to write OEP code. My special thanks to Tatyana and Thomas Albert for help with the
preparation of the dissertation text.
However, there are some other pertinent considerations before doing so. The
condition that the KS single determinant gives the exact density through first order
in its correlation perturbation, obviously imposes the condition that p' 0. This
density condition is very general as it applies to exchange and correlation, and has
been used quite successfully in defining ab initio dft exchange and, in higher orders,
correlation potentials[13, 16, 17, 48], since it also transcends any particular separation of
the Hamiltonian in perturbation theory. In particular, it is not subject to HKS = hs(i),
and thereby alleviates the pathological behavior that gives no convergence when defining
such correlation potentials from MBPT(2), for example [13]. For the exchange only,
however, such effects are less important[16, 17]. Therefore this paper will use the sum of
Sham Hamiltonians, but the principle for derivation remains the same.
Hence, requiring that
0 PKS (KS ARV \KS) + (KKS VR0 I KS) (511)
A p(x, x ) (512)
a \ + lab Wab\
Rl C Ca) 1 + 2 )(CtI + C Ca Cb) ( +... (513)
i,a I
Ih)R(hl + h2)Rl(h2 +... (514)
V= [ (i)+ (i) (515)
i
Single excitations in the resolvent operator are indicated by a) ,and collectively by the row
matrix, hi),doubles by b), h2),etc. It is easy to show[17] that only the singles contribute
to the firstorder density, and
Ps y i I )fia +c 2 k(1l)a(1)iK + xa)/(i )a) 0 (516)
?,a i,a
Then this condition defines the Vx(1) operator up to a constant[52]. Note that this
is a pointwise condition as it should equal zero at all xi, but actually imposes the weaker
exchange,
7013 P13
K(1) (1) 3) 3) 1) (3) r1tp(3)d3j(l1) (52)
J J
where the {Qj}are the occupied KohnSham spin orbitals, p=i,j,k,l.... The latter are the
solutions to the equations,
..,/( (1) CpOP(1) (53)
hsl') = h(1) + J(1) + Vx(l) (54)
Ji() E I (2) 1 (2)d2 (55)
p(t) d2 (56)
I 12
PKS () i () (57)
where the unoccupied orbitals are indicated by p=a,b,c,d....
The condition that the density, p(l), be the HartreeFock one, is that
l, (HFlh+ J K aHF) 0 (58)
PHF(l) I *( (1) (5 9)
which, as is wellknown, is correct through first order in correlation measured relative to
the sum of HF oneparticle operators, Ho = Y f(i), due to the MoellerPlesset theorem.
For KS orbitals, fia is not zero. Hence, if the objective were to maximize the similarity
in the KS and HF density, it would then require minimizing a quantity composed of KS
orbitals, related to
min \(if 1sa)2 = min1 (iK + x a)2 (510)
i,a i,a
,.I :i.; a kind of leastsquares minimization as has been discussed previously[13, 16, 17,
48]. Clearly, this minimization would have the trivial solution that (i K a) = (i Vx a).
a(r4 r s5 r6 6(r4 r 5 r 6) 6(r4 r5 r 6)4 8 (r4\ r\ r) 6(r 4 r 6)6b
r4 r r6(r 4 rs\ r 6) 4 r56 r 6)
k a 6(r4 rs5 r6)l 6 ) rrr) r5 r 66 )jr r 6 r4 r5 5 r 6
4 5\ r6 )k k k
(SX10) (SX11) (SX12) (SX13) (SX146 22)
Interpretation of all diagrams which contributes into the second kernel and details of the
implementation are given in Appendix B.
6.1.3 Properties of the Second ExactExchange Kernel
All the diagrams contributing to the second exactexchange kernel have only one
contour and because of this fact only aaa and 333 spin components are present, as
in Goldstone diagrams where Sz has to be conserved at each vertex. The algebraic
expressions for all the diagrams have to be symmetric with respect to permutations of
rl, r2, and r3 variables. In the case of the twoelectron spinunpolarized systems the OEP
exchange potential becomes equivalent to the Slater potential[38]
vSlater(r) _ i()j() r l) l (623)
Sizj PA IJ r ri(
Taking into account the HOMO condition and considering the expression for the exchange
kernel, it is possible to write for the special case of two electrons
,f EP(p1, 2) (624)
ri r2
Diagrams (VX1)(VX5) and (VX8) of (620) will cancel diagrams (SX7), (SX8) and
(SX11)(SX14) of (621) and (622) because of the HOMO condition and the behavior of
the HartreeFock exchange for the case of two electrons. Diagrams (VX6) and (VX7) will
cancel diagrams (SX1) and (SX2) because of Eqn (623). Diagrams (SX5), (SX6), (SX9)
and (SX10) will cancel diagrams (F1)(F4), (F7)(F10) and (F13)(F16) because of Eqn
(624). Finally the sum of the diagrams (SX3) and (SX4) cancel the sum of (F5), (F6),
(Fll), (F12), (F17) and (F18). Thus for the special case of two electrons
gOEP (12, 3) 0 (625)
The definition of the exchange part of E,, can take into account its definition in
wavefunction theory,
E, =< ,IV, > EH (138)
KohnSham orbitals are defined by an effective local potential vs. Transforming the
variational condition on the energy functional into a condition for the constrained search
over the orbitals, it is possible to write
(r) (E[p] T,) 6(E.Et +EH + Exc)
vs(r) vext + VH + Vc (139)
6p 6p
where the exchangecorrelation potential is defined as the functional derivative of the
exchangecorrelation energy
c(r) 6 (140)
6p(r)
Once Exc is approximated and the KohnSham equations are solved, the total energy can
be found from the following expression
E = E drldr2P(1)P(2) drprlxc() + E (141)
i= 1 2 Iri r2
The solution of the KohnSham equations is completely analogous to the solution for
the HartreeFock case, and it is usually done by an SCF procedure. After selfconsistency
is reached, the KohnSham orbitals are guaranteed to reproduce the true density of the
]rn i,electron system.
Virtually all modern implementations of DFT use the KohnSham scheme. However,
the theory still has open questions as to how to construct the exchangecorrelation
functional. Thus, the main challenge for the theoretical development of DFT remains the
construction of accurate exchangecorrelation functionals.
1.2.1 TimeDependent Density Functional Theory
The KohnShame scheme provides the possibility to describe ground state energies
and densities. For the description of excitation energies a timedependent generalization of
conventional DFT can be used. Ordinary timedependent DFT is based on the existence
(VC2 9) + (VC2 10)
4 < ablij >< ijlkb > (< ckA >< acp > + < acA >< ck/ >)
(E, E, Ea + Eb) (Ek E) (Ek Ea)
(VC2 11) + (VC4 7)
4 < ablij >< ijlkb > (< ckA >< acp > + < acA >< ck/ >)
b,c,,j,k (cE + j a  b)(i + Cj Eb Ec)(Ek Ec)
(VC2 12) + (VC3 8)
4 < ablij >< ljlkb > (< akA >< ili > + < ilA >< akp >)
,b,i,,, (i + j Ea b)(El + Cj Ea Eb)(Ek Ea)
(VC2 13) + (VC2 14)
4 < ablij >< ijlkb > (< alA >< lki > + < alp >< lkA >)
a,b,i,j,k,l ( + a b)(k a)( a)
(VC2 16)
< ablij >< ijlcb >< akA >< ckp >
a,b,c,i,j,k + a Eb)E E) k c)
< ablik >< ijlcb
(Ei + EF 
< ablik >< ijlcb
(Ei + k a 
(VC3 3) + (VC3 5)
> (< jkA >< ili > + < jkp >< ilA >)
*b)(Ei + k Ea Eb)(El + j E Eb)
(VC3 4) + (VC4 5) =
> (< jkA >< ac/ > + < acA >< jki >)
Eb)(Ei + k a b)(Ei + j b Ec)
(VC3 6) + (VC4 3)
> (< jkA >< ac/ > + < acA >< jk/ >)
Eb) (i + Ek Ec Eb) (i + j Ec Eb)
(A22)
(A23)
(A24)
(A25)
(A26)
S< ablik >< ljlab
a,b,i,j,k,l ( + 
a,b,c,i,j,k
a,b,c,i,j,k
(A27)
(A28)
(A29)
In this way, there is no dependence on the virtual orbitals as is deemed computationally
or formally important in some DFT circles[59] Of course, using the usual finite basis set
computational tools of quantum chemistry, the distinction is less important. This formula
has recently been applied[58].
Because the OEP equation is a pointwise identity having to be zero formally for all
x in a finite basis set it corresponds to a :r, ii I.,few mapping in the general case. The
dependence upon the delta function makes the OEP procedure sensitive to the auxiliary
basis set used in the calculation. However, the SVD procedure handles the null space
functions plus any potential linear dependency in the basis set in a fairly automatic
v ,i[;n]. If the X matrix is removed, this is not the case, and some algorithms can result
in unphysical or trivial solutions under certain combinations of auxiliary and molecular
bases. C'!.... i to invoke the average energy denominator and the resolution of the
identity A'. f. the leastsquares mminimization, will not give Eqn(24). However, once
we eliminate the delta function that makes the approximation pointwise, we reduce the
problem to the minimization of the variance
min Var min (i(K + Vx)2i) j iK + Vxi)(jlK + Vxli) (534)
i i,j
that can be used to define a Vx but one that is not pointwise, and, consequently, does
not satisfy all the conditions above. This further simplification of the weighted least
squares approach in Eqn (526) has also been considered recently [58]. Ultimately, one
primary criteria for the best exchangeonly potential should be the satisfaction of the
Janak theorem. As shown in Table 51, and pointed out previously[16, 48] this is difficult
to achieve in any normal basis set.
3.2 Kernel for the SecondOrder Optimized Effective
Perturbation Theory Correlation Potential .
3.3 Properties of the Correlation Kernel .......
3.4 Numerical Testing .. ..............
3.5 Conclusions . . . . .
Potential ManyBody
4 AB INITIO DENSITY FUNCTIONAL THEORY FOR SPINPOLARIZED
SYSTE M S .... ................... ..............
4.1 Theory ................... ..............
4.2 Results and Discussion .. .....................
4.2.1 Total Energies . . . . . . .
4.2.2 Ionization Potentials .. ..................
4.2.3 Dissociation Energies .. ..................
4.2.4 SingletTriplet Separation in Methylene .............
4.3 C conclusions . . . . . . . .
5 EXACTEXCHANGE TIMEDEPENDENT DENSITY
ORY FOR OPENSHELL SYSTEMST ..........
5.1 ExactExchange Density Functional Theory .
5.2 TimeDependent Optimized Effective Potential ..
5.2.1 Theory and Implementation ........
5.2.2 Numerical Results ..............
5.2.3 C(!i igeTransfer Excited States ......
5.3 Conclusions . . . . . .
6 EXACT EXCHANGE TIMEDEPENDENT DENSITY
ORY FOR HYPERPOLARIZABILITIES .. .....
FUNCTIONAL THE
..............
FUNCTIONAL THE
..............
6.1 T heory . . . . . . . . .
6.1.1 TimeDependent Density Functional Theory Response Properties
6.1.2 Diagrammatic Derivation of the Second ExactExchange Kernel
6.1.3 Properties of the Second ExactExchange Kernel .. .......
6.2 Num erical Results . . . . . . . .
6.3 C conclusions . . . . . . . . .
APPENDIX
A INTERPRETATION OF DIAGRAMS OF THE SECONDORDER MANY
BODY PERTURBATION THEORY OPTIMIZED EFFECTIVE POTENTIAL
CORRELATION KERNEL .. ........................
B INTERPRETATION OF DIAGRAMS OF EXACTEXCHANGE SECOND
K E R N E L . . . . . . . . . .
REFERENCES ......................................
with the PBE potential are numerically close, and close to the reference CCSD(T)
energies. This, of course, is part of the power of GGA and meta DFT, even when such
energies are obtained with highly erroneous potentials. Instead, PBE partly benefits from
the cancellation of errors in V, and V, (Fig 1 and 2). For most of the openshell systems
the energies of the semicanonical OEPMBPT(2) are much closer to CCSD(T) then the
corresponding PBE energies, which is also a consequence of the wrong behavior of the
PBE potential.
4.2.2 Ionization Potentials
Orbital energies and the corresponding ionization potentials are much more sen
sitive to the exchangecorrelation potential than the total energies. Vertical ionization
potentials, calculated as energy differences between the neutral and its ionized system are
calculated using the same basis sets and geometries as in the previous section. Results
are presented in Table 42. In exact DFT, the energy of the highest occupied molecular
Table 42. Ionization potentials (in e. v.)
HOMO, OEPsemi AE, OEPsemi HOMO, PBE AE, PBE Exp
Ne 21.01 21.59 13.35 21.69 21.56
N2 16.89 15.03 10.27 15.41 15.58
CO 13.68 13.69 9.05 13.87 14.01
CN 4.19 3.51 0.149 3.72 3.86
H20 12.37 12.73 7.08 12.46 12.62
N 15.25 14.52 8.30 14.73 14.53
Li 4.89 5.38 3.23 5.59 5.39
orbital (HOMO) corresponds to the negative of the exact vertical ionization potential. In
the case of the semicanonical OEP the HOMO energy is close to the corresponding AE
values, what can be explained by the correct shape of the exchangecorrelation potential
(Fig 41 and 42 and ref [16, 17]), while PBE does not exhibit correct behavior of potential
and thus, fails to reproduce the correct HOMO energy values. Failure to reproduce the
correct HOMO energy causes the incorrect integer discontinuity of the PBE (or any other
GGA) exchangecorrelation potential[39, 40].
the exchange and correlation potentials as a functional derivative of E" + E(2) [13, 33].
However, this is a numerically hopeless procedure without any resummation of terms: (1)
it suffers from adding a large diagonal term into the perturbation; (2) MBPT(2) is not
(2)
bounded from below and any effort to use the variational condition 6E is likely to suffer
from numerical problems.
Our approach is to correct the above by changing the choice of Ho and focusing on
the expectation of the density operator instead of any variational functional derivation.
The partitioning of the hamiltonian is chosen to be the onebody part as the zerothorder
approximation
a, 3
Ho0=^ fiI{aIa+, } + Y fbbatabh} + fij{aaa} + f abab aataa} (46)
a i b i'j a4b
However, to avoid dealing with nondiagonal Ho, a semicanonical transformation will be
performed to obtain the more convenient zerothorder hamiltonian,
a,l3
Ho fpp{ ap} (4 7)
( p
Dropping the ~ for simplicity, with the semicanonical partitioning of the hamiltonian the
OEP exchange and correlation potentials assume the following form
V(r) 2 dr1 < ijlja > i(rl)a(r)( l(, (48)
Sijc fiir faa 1(
a,3
V, f Jdr l(25 < 1,jTlabr > (< aUbTlc,jT > < aobrljTc, >)(Pi(ril)(Pco(rl)
i,j,abc( + fjj faac fbbr)(fia fc)
S i,j,a,b,3
_ K < ijTa UbT > (< ijlkbT > < jiklb >)(Pkc(Fl)oPao(1)
2y Y < > (< >
Si,j,k,a,b iia jjf r faaa fbbT)(fkkc faaa)
, K < ij, iab > (< aUbIkj, > < braUIkj >)*kaC. (1)i(r1)
Sijk,,b fii ( + fjj faaa fbbr)(fkk + fjjr faaa fbbr)
quite stable, and does not suffer from the problems alluded to by Staroverov, et.al, even
when the auxiliary basis has a dimension greater than the MO basis.
In another interesting example, Rohr, et al[49] observes the occurrence of a degen
eracy in performing OEP calculations, and to investigate this, intentionally construct a
problem where the HOMO and LUMO orbitals of the KS problem are degenerate. Subject
to direct minimization, they proceed to report on the failure of the OEP procedure. On
the other hand, it is shown in this paper that the original X based approach handles this
problem, too.
The exchangeonly timedependent OEP has been considered by Goerling[20] and
implemented by Hirata et. al[21, 50]. However, all results to date are for closedshell
systems. To further address the exact local (OEPx) exchange versus timedependent
HartreeFock (TDHF), TDOEPx is generalized to treat excited states for openshell
species. Results from adiabatic TDOEPx are in good agreement with TDHF for both
excitation energies and polarizabilities, however, it is shown that chargetransfer states
cannot be properly described in TDDFT. For standard DFT methods, gradientcorrected,
hybrid, etc. this has been noted, and improved upon[51], but standard methods, unlike
TDOEPx, still suffer from other limitations, like the selfinteraction error and the incorrect
longrange behavior of the potentials and kernels. In an approach that gives the 'right
answer for the right reason' these exact conditions are requisite. In TDOEPx there are no
such errors. Hence, failures of TDOEPx compared to TDHF have to be exclusively due to
the local versus nonlocal exchange operator, which is demonstrated.
Finally, C6 coefficients are obtained for openshell systems from the TDOEPx
freu iiv. dependent polarizability.
5.1 ExactExchange Density Functional Theory
The most satisfactory way to introduce exact exchange DFT is to insist that
the KohnSham single determinant provide the exact density, but subject to a local,
multiplicative exchange operator, Vx(1), instead of the usual nonlocal HartreeFock
When we need to calculate polarizabilities at frequency u, the following system of
linear equations (540) should be solved
(A A B d" h
h) (540)
B A + ul d`+ h
where
hpq = / dr(pp,(r)ipyq,(r) (541)
Once these equations are solved for d, the dynamic polarizability is readily evaluated from
a, 3
a(w; +w) 1 Z (h jh d + h,,d) (5 42)
i,a a
For the calculation of the C6 van der Waals coefficients, the system of equations (540)
should be solved for the imaginary frequencies iu from which the coefficients can by
calculated by
3 r[
C/0 \ dwa(iw)a(iw) (543)
T Jo
where a(iw) = ((awxx ) + ,yy(iw) + a (iL)). Integration of equation (543) was carried
out by Gauss('l. li!i. v quadrature.
To ensure the correct ..i mptotic behavior of the exchange potential, the
ColleN. bet algorithm[30] was used. As a seed potential, we used the one proposed by
,i ,, i [: ]
V later(T) jo dr (5 44)
,j P, Ir rll
For the spinpolarized case two different Slater potentials are used for the corresponding
spin component of the local exchange.
5.2.2 Numerical Results
Excitation energies and polarizabilities are calculated for several openshell systems.
Equilibrium geometries are taken from ref[36]. For all calculations the uncontracted
CHAPTER 2
INTERCONNECTION BETWEEN FUNCTIONAL DERIVATIVE AND EFFECTIVE
OPERATOR APPROACHES TO AB INITIO DENSITY FUNCTIONAL THEORY
The use of the KohnSham orbitals and orbital energies to construct an implicit
densitydependent energy functional using,e.g perturbation theory[12, 13] is a straightfor
ward approach to the ab initio DFT. The corresponding KohnSham potential can then be
obtained by taking the functional derivative of the finiteorder energy via the chain rule,
that transforms the derivative with respect to the density into the derivative with respect
to orbitals and orbital energies[14].This leads to the Optimized Effective Potential[13, 15]
equations which can be complicated already in second order[13]
Alternatively, one can determine the first and second order KohnSham potential
by requiring that the corresponding first and second order perturbative corrections to the
reference density vanish.Unlike the functional derivative approach such a condition on
the density can be described with standard rii ii:body techniques.The recent work[17]
uses diagrams to derive the secondorder OEP equation in a systematic and compact
fashion,while a second paper [16] does so algebraically.
Both the functional derivative and density(effective operator) approaches lead to
exactly the same equation in the first order[17].However,the functional derivative of the
second energy involves a certain type of denominators that are not present in the density
condition(effective operator approach).Still the terms involving such denominators can be
transformed to match the densitybased equation exactly. This caveat raises the question
of whether the two approaches are equivalent in higher orders,or for different partitionings
of the hamiltonian.
2.1 Equations for the ExchangeCorrelation Potential in the Functional
Derivative Approach
To derive the equations in both approaches we first split the full electronic hamilto
nian into a perturbation and a zeroorder hamiltonian:
H Ho + V (21)
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CHAPTER 3
AB INITIO TIMEDEPENDENT DENSITY FUNCTIONAL THEORY EMPLOYING
SECONDORDER MANYBODY PERTURBATION THEORY OPTIMIZED
EFFECTIVE POTENTIAL
The timedependent OEP exchangeonly method in the adiabatic approximation
has been previously derived [20] and implemented for molecules by Hirata et. al [21].
Numerical results in that work show a reasonable description of both valence and Rydberg
excited states, partly due to the correct .,i!!, ,l'tic behavior of the exchange potential,
but also due to the elimination of the selfinteraction error. In particular, OEPTDDFT
is superior to standard approaches like TDDFT based upon local density approximation
(LDA) or BeckeLeeYangParr (BLYP) functionals. Similarly, exchangeonly OEP with
exact local exchange (EXX)[22, 23] has been shown to greatly improve bandgaps in
polymers [23]
Another advantage of OEP based methods is that since virtual orbitals in the
exchangeonly DFT as well as occupied orbitals are generated by a local potential,
which corresponds to the Nparticle system, the differences between orbital energies of
virtual and occupied orbitals offer a good zerothorder approximation to the excitation
energies[16, 17].This is not possible in the case of HartreeFock theory, where occupied
orbitals are generated by an Nl particle potential and the energies of unoccupied orbitals
come from Nelectron potential, and thereby, approximate electron affinities.
Once OEP correlation is added [13, 17], the essential new element in the time
dependent DFT scheme is the exchangecorrelation kernel, which in the adiabatic approx
imation is defined as a functional derivative of the exchangecorrelation potential with
respect to the d, i,il [20]:
6p(r2)
Hence, in this paper we will derive the kernel for the OEPMBPT(2) correlation potential,
when we use the standard KS Ho, Ho = i hks(i). In prior papers[16, 17] this has
been called OEP KohnSham (KS) to distinguish it from other choices for Ho. It is
The matrix elements Fq, and e0, can be calculated from the relations (6 6)
F\ = h (AP + BRPq) UrA
q, pq\aiu iq a
a,i
6.1.2 Diagrammatic Derivation of the Second ExactExchange Kernel
For the derivation of the second kernel it is convenient to use the form (68)
g9(ri, r2, r3)
6fxc(r, r2)
6p(r3)
Which can be rewritten as
f(r,=r2) dr4dr5 6X (rr4) Qr 5)X (r5,r2)+
Sdr4dr5X (rI r4) 6Q(4, r5) X (r5, 2)
Jdr4dr5X l(rr) (4,r5) 6p(r3)
Taking into account the fact that
6X\(r^_, r2)6X(rs, r4 _
(rl r2) dr4drf X (r, r) ( 4)X '(r2, 4)
bp(r3) V,(rs)
and doing some simple transformations, it is possible to write
g(rl, r2, r3)
drdr7dr8sf(r2, rs8)X (r3,6) X(r, r) X (r, r7) +
J 6Vs,(r6)
dr4dr5dr6X(rl, r4)X (r3, r6)Q(4,r5)X (r2,r5)
r ir (Vs(r6)
dr6dr7dr8f (ri, r7)X (r3, r6) 6X(r, r) X(r2, 8)
j 6bV,(r6)
(611)
where
Sdr' p() + V(r)
(66)
(6 7)
(68)
(69)
(610)
Vs(r) = ve.t(r) +
(612)
of selfinteraction is assured. The correct .i'!.' ictic behavior is achieved using the
ColleN. bet algorithm[30].
When we solve the TDDFT equations with the kernel developed in this paper,
we obtain the results for excitation energies shown in Table 32. In general, just as we
observed from the orbital energy differences, the OEPMBPT(2) results tend to fall on the
low side of experiment and the OEPx on the high side. In fact, except for the excitation
energy for the lowest 1P state, where the very low orbital energy of 7.58 biases the
results, an average of the two would seem to be about right. There are, however, still
difficiences in the basis set, as seen by the EOMCCSD results compared to experiment,
where besides the additional correlation effects introduced by EOMCCSDT[31], the
further extension of the diffuse functions would remove the 1 eV error in the highest lying
(1S) state. Of course, the dependence of TDDFT on the basis and that for a twoparticle
theory like EOMCC should be quite different.
The great sensitivity of the results to the orbital energies from the underlying KS
DFT calculation can be further appreciated by simply taking the energies from the OEPx
results and using them in the evaluation of the kernel and the matrix elements in the
TDDFT equations. These results are shown in Table 33. The near coincidence of results
for OEPx and OEPMBPT(2) is apparent, with both now being too high.
A similar experiment can be made where we use orbitals and orbital energies from
OEP2(sc) ab into dft results as described elsewhere[16], to obtain the results in Table
34. That is the TDDFT equations and the kernel are assumed to be the same, but we
use orbitals and orbital energies obtained from OEP2(sc). Here we also show the results
from standard TDDFT applications using the LDA and B3LYP functionals for comparison
purposes. Clearly, we have improved results at both the OEPx(sc) and OEP2(sc) levels,
with the same pattern of the former being too high, but less so; and the latter, too low,
but better than before. It is apparent that the proper way to achieve the benefits of
At t = to the perturbation is applied so that the total potential is given by
v(r,t) '= (r) + vi(r,t) (153)
where vi(r,t)=0 for t < to.
The objective is to calculate the linear density response pi(r, t) to the perturbation
vi(r, t). Conventionally, pi(r, t) is computed from the full linear response function X as
pi(r,t) dr dt'x(r, t,r',t')vi(r',t') (154)
Since the timedependent KohnSham equations (150) provide a formally exact way of
calculating the timedependent density, it is possible to compute exact density response
pi(r, t) as the response of the noninteracting system
p (r, t) = dr' dtt'XKS (T,, ', r t1) (155)
where v~i) (r, t) is the effective timedependent potential evaluated to first order in the
perturbing potential, i.e.,
v) (r, t) = v, (r, t) + dr'p t')+ + dr! f dt'f,(r,(t,r', t')pl(r',, t') (156)
The exchangecorrelation kernel fc, is given by the functional derivative of v,,
f(r, r', t') v[P]) (157)
p(r',P')
at p = po.
While the full response function X is very hard to calculate, the noninteracting
XKS can be computed fairly easily. In terms of the static KohnSham orbitals the Fourier
transform of XKS(, t, T t') with respect to (t t') can be expressed as
OCC 11OCC
cKS ( r) W W1 ()58)
XKS(F, r', w) limg a w a (158)
St( a (i ,
Table 56. Static polarizabilities (in a. u.)
TDHF TDOEP SVWN CCSD
CN
Xyy
COzz
a
Aa
ayy
co+
azz
a
Aa
CH3
ayy
azz
a
Aa
N
11.98
11.98
18.55
14.17
6.57
6.66
6.66
12.22
8.52
5.54
15.06
14.04
15.06
14.72
1.02
11.91
11.91
18.41
14.08
6.50
6.66
6.66
12.15
8.49
5.50
15.00
13.98
15.00
14.66
1.02
15.63
15.63
25.00
18.75
9.38
9.38
9.38
14.06
10.94
4.69
18.75
18.75
18.75
18.75
0.00
12.50
12.50
23.42
16.16
10.94
9.38
9.38
12.50
10.42
3.13
15.63
15.63
15.63
15.63
0.00
aZ 6.24
Table 57. Isotropic (
TDH
CN ... CN 93.A
CH3 ... CH3 94.:
N ... N 19.:
6.23 6.25 5.94
76 coefficients (in a. u.)
[F TDOEP
37 92.06
27 93.40
28 19.21
ROOSADZP basis set[27] has been used. Results for excitation energies are presented in
Table 53.
Table 53. Excitation energies (V valence state, R Rydberg)
TDHF OEP EP TDOEP SVWN CCSD Exp
CN
2I(V) 4.18 8.16 4.91 1.96 1.52 1.32
2Z+(V) 5.47 10.58 5.48 3.19 3.62 3.22
CO+
2II(V) 7.55 9.98 8.23 3.03 3.43 3.26
2Z+(V) 11.14 9.37 10.82 4.99 6.14 5.82
CH3
2A' (R) 6.52 6.53 6.42 5.00 5.88 5.72
2A'2 (R) 7.93 7.97 7.93 5.93 7.18 7.44
N
4P (V) 9.84 11.11 9.88 10.98 10.84 10.35
4P (R) 13.59 13.16 13.18 11.76 1 13.62
As follows from the first two columns of Table 53, TDHF and TDOEPx produce
results of approximately the same accuracy, as would be expected by the fact that the
local exchange in TDOEP is meant to be a kind of leastsquares fit to the HF nonlocal
exchange potential. This certainly makes the occupied orbitals quite similar, but the
spectrum of unoccupied orbital eigenvalues is very different, as shown in Table 54, since
TDOEPx will generate a Rydberg type series instead of anything like the HF virtual.
As is well known, the latter are determined in a potential of nelectrons, making them
appropriate for electron attached states, while the occupied ones feel a potential of n
1 electrons. To the contrary, the orbitals obtained in OEPx have the same potential
for an electron in the occupied and unoccupied orbitals, which is why the latter more
nearly simulate Rydberg states, as some of the unoccupied orbitals will have negative
orbital energies. Of course, in a finite basis as long as the space separately spanned by
the occupied and the unoccupied orbitals is the same, there would be no difference in the
results, though the diagonal values (but not the trace) would change. The significant
1 This Rydberg state was not obtained in the EOM CCSD Davidson diagonalization
(VC12 10)
S fibfbj(< iaA >< jap >
(Eb Eb)(E~ Ea)(E Ea)
i,j,a,b
(VC12 11) + (VC15 6)
4 fib < aclbj > (< iaA >< cjp > + < cjA >< iap >)
ab,c,i,j (Ei Eb)(EFi Ea) (E Ec)
(VC12 12) + (VC15 1)
4 E fab < bclij > (< iaA >< cjp > + < cjA >< iap >)
a(Eij ( ~b) (Ei Fa) (E' Ec)
2 fab < jb\iC
a,b,c,i,j
2 fab < cblij
a,b,c,i,j
2 fia < b a
a,b,c,i,j
Sb fia < bcja
a,b,c,i,j
(VC12 13) + (VC13 
iaA >< cjpi > + < cjA ><
 Eb)(Ei Ea)(Ej Ec)
(VC12 14) + (VC14
iaA >< cjpi > + < cjA ><
 Eb) (Ei Ea)(j Ec)
(VC12 15) + (VC14 
ibA >< cji > + < cjA ><
 Eb)(Ei Ea)(Ej Ec)
(VC12 16) + (VC13
ibA >< cjp > + < cjA ><
 Eb)(Ei Ea)(Ej Ec)
12)=
iap >)
1)
iap >)
10)
ibp >)
8)
ibp >)
(VC13 1) + (VC13 2) =
S fai < bilaj > (< bkA >< kjp > + < kjA >< bkp >)
i,j,k,a,b (i a)(E Eb)(E Eb)
(A105)
(A106)
(A107)
(A108)
(A109)
(A102)
(A103)
(A104)
> (<
(Ei
> (<
(Ei
(<
(Ei
(<
(Ei
and frequently diverges. Since the density condition approach is completely equivalent [37]
to the variational OEPMBPT(2) for the KS partitioning it has the same problems.
However, in the case of the nonvariational, semicanonical density condition approach, the
unboundness from below is less of a problem. The semicanonical choice of Ho provides
good approximations to the energy and wave function in MBPT(2) and the corresponding
total energies are much closer to highlyaccurate CCSD(T) ones, at least in the chosen
basis. The computational cost of the OEPMBPT(2) method is comparable with the cost
of MBPT(2), the scaling of both methods is N5, where N is the number of basis functions.
This is more expensive than conventional DFT (scales like N3), but less expensive then
CCSD(T), computational time of which is proportional to N7. The correlation potential,
Table 41. Total energies
MP2 OEPKS OEPsemi PBE CCSD(T)
N2 109.449149 109.7,11 11, 109.457779 109.457071 109.468618
N+ 108. :. ;;, no conv 108.905444 108.890773 108.900556
N 54.544740 54.593111 54.545199 54.535569 54.5 !,4
H20 76.370003 76.510744 76.373092 76.369991 76 ;"7
H20+ 75.901155 75.989525 75.902958 75.912146 75.921765
CN 92.598196 no cony 92.651919 92.646898 92.658287
CN 92.772714 no cony 92.780718 92.783715 92.796498
CO 113.228520 113.511203 113.237817 113.239593 113.251239
CO+ 112.702208 no conv 112.734813 112.729947 112.739892
02 150.205776 no conv 150.221870 150.255051 150.221560
Ne 128.859598 128.951437 128.861175 12.> ..844 1 '. ,133
Ne+ 128.066796 no conv 128.067574 128.068711 1 NI IIII'
Mean error, 58.22 26.74 36.16
kJ/mol
calculated with the semicanonical OEP for the Ne atom[16] is close to the corresponding
quantum MonteCarlo potential, and we can expect that for other systems, where QMC is
not available, semicanonical OEPMBPT(2) potentials should be a good alternative. For
the openshell case potentials produced by KohnSham OEP show an overestimation of
the correlation energy, the same situation previously reported for the closedshell case[16].
(Fig 41 and 42). The correct 1 longrange ..imptotic behavior of the OEPMBPT(2)
The HartreeFock exchange functional provides an exchange part of orbital
dependent exchangecorrelation potentials. KohnSham DFT with OEP exchange
potentials provides a superior results to all known densitydependent exchange func
tionals. To make OEP potentials useful for practical calculations it is necessary to have
the correlation potential, which can be combined with it. Conventional GGA correlation
functionals are developed together with the corresponding exchange potentials, which are
very inaccurate. Because of this correlation potential of GGA compensates errors in the
exchange part. In most cases GGA correlation potentials have an opposite sign to a nearly
exact QMC potentials. The correcting terms incorporated into correlation potentials,
so it is really difficult to extract the "real correlation p oii So far, it is not surprising
that combining GGA correlation potentials with OEP exchange potentials the associated
results are inferior to exchangeonly cases. Since conventional correlation functionals are
not compatible with OEP, the development of orbitaldependent correlation functionals is
necessary.
Ab initio DFT uses orbitaldependent correlation energy functionals from the
rigorous wavefunction methods. Since corresponding correlation potentials are derived
from ab initio methods, they are compatible with OEP exchange potentials. Unlike
conventional ones, ab initio functionals are systematically improvable, since one can ahi,
use a higherlevel approximation to obtain a more accurate functional. They also have a
welldefined limit represented by the FCI method.
1.2.5 Ab Initio Density Functional Theory
The main advantage of orbitaldependent energy functionals is the fact, that
such functionals are known in analytical form. Particularly, the exchange functional can be
written in following way
E < Vee >EH < ;.;,; > (167)
i,j
The expression, which is suppose be differentiated with respect to the potential on the
righthand side of equation (310) can be represented by diagram (V) (311)
i a
V (n)
xc
(V) (311)
The functional derivatives from diagram (V) produce the set of diagrams (V1)(V6) of set
(312)
16 6 2 6 1 6 2 6
b 1 b 2 1 2b
I a (n
S 2 1 a 2 V a V_ I
V a V (n) a V (n) I V (n
xc xc xc xc
(VI) (V2) (V3) (V4) (V5) (V6) (312)
During the derivation of diagrams (V1)(V4), diagrams containing "brackettype" denom
inators will appear, because one of the steps in the diagrammatic rules of differentiation
requires detaching the unoccupied line from diagram V and inserting the last two parts of
the 6 function (226). Such diagrams can be transformed into a set of regular diagrams,
using the diagrammatic relation (313). This relation shows us that for the transformation
to regular diagrams it is necessary to double each of the denominator lines and insert the
last two members of the 6function (226), subject to the restricted summation, I / k.We
use the same procedure for the case of occupied orbitals.
Sa > 1 1
2 2 2> 2
k 2 > k k
+ :k + + + (Dl)
m .
(313)
UHF
...... MBPT(2)
 OEPsemi
 CCSD
 CCSD(T)
1.0 1.5 2.0
R,A
Figure 45. HF potential energy curve.
99.3 
99.4 
99.5 
99.6 
99.7 
99.8 
99.9 
100.0
100.1
100.2
100.3
100.4
2.5
2.5
two diagrams gives the diagrams of equation (234)
Si j J
2 +2 (234)
On the two diagrams we impose the restriction i / k. To make a direct correspondence
with diagrams 7 and 11 we need to add diagrams that arise from taking functional
derivatives from denominators (235)
k b j k a b /
2 2 (235)
After these diagrams are added to the previous two, we will have a direct correspondence
with diagrams 7 and 11. When functional derivatives are taken with respect to the
unoccupied orbitals and an unoccupied line inserted, the same procedure will give a direct
correspondence with diagrams 6 and 10 of equation (225).
Now consider the first diagram for the secondorder energy expression. The pro
cedure for taking functional derivatives from the right side of the vertex, which do not
contain a closed Fring are absolutely the same as in the case of other second and third
diagrams in the energy expression. When such functional derivatives are taken, we have
four diagrams corresponding to diagrams 1215. To take functional derivatives from the
Frings, the first diagram in the energy expression must be represented with more detail,
taking into account that f = h + J K,in following the form (236)
CF X + X (2 3
F X+ )+ + 2 X +2 +2 (236)
) 2007 Denis Bokhan
tb2 b)< ik\j b > *F9
i,j,k,a,b F b) 
F9 = (< ,,,, i.> >< pbpjrl >< ia > + < ',, i.K >< pbpj >< Cjac] > +
< ,, > >< b~ >< > + < ', r, q >< b(6jK >< (OiOa > +
< ' >< j>< b( >< (pi(p > + < 'r, r,7 q >< ib Pj >< (ri(baK >) (B16)
D12 2Z (< Papc \bcj > *F10
D12 2
(Ei Ec) E( Eb (Ei Ea)
i,j,a,b,c c)( b)(i a)
F10 (< >, ,v. >< cpipr] >< Cicga( > + < ><, .. >< ci< c >< aGil > +
< .< >< OcrX >< aOiaK > + < >' < ,* K>< c >< a( i(a> +
< , , >< cj >< ( i(al > + < ' 17 >< ><< >( i(aK >) (B17)
D13 2 << =ba > *F11
D %:(E Ec) E_ Eb_)(i_ E)
i,j,a,b,c ( i
F11 = (< cp >< >< bj( > + < cK ac( >< > bjl > +
< iPc( >< OaOc)l >< Vb(PjK> + < icl >< acK b>< Vj > +
< i< acK >< Xb~Pj1l > + < Ol >< ac( >< Vb(jK >) (B18)
D14 2 > < , ,.'~ ba > *F12
ijkab k Ea) (E Eb)(Ei Ea)
F12 = (< ickKX >< cackl >< ,, .< >+< kK Ci~ca>< O c k >< ',.>+
< ik( Cl > < ',OaOk .i.' > + < OOkl >< OaOk > 'K ..S > +
< >ik( >< OaOkK >< ,. > + < cick X >< Oa(Ok >< ' . >) (B19)
To get final expression for BK all terms D1 D14 should be summed up.
After the transformation of irregular diagrams and adding the functional derivatives from
the denominator of diagram (V), diagrams (V1)(V4) will appear. Using diagrammatic
expressions for the first member of equation (39) and diagrams (V1)(V6) it is possible to
build the exchangecorrelation kernels to any order.
3.1.2 An Example: Diagrammatic Derivation of ExchangeOnly Kernel
The kernel for the exchangeonly case was initial derived by G6rling [20], and then
rederived and initial implemented by Hirata et. al [21]. Such a derivation requires much
effort. Here we offer a facile derivation with the diagrammatic formalism.
The exchange energy (Ex=~E~ j < ij ji >) can be represented by diagram (314).
E, (EX) (314)
After taking the first functional derivative with respect to V we will have the diagram
(315).
a
E,/6V, = 2
(VX) (3 15)
After that the diagrammatic rules must be applied one more time to get the second
functional derivative, as is necessary according to equation (39). After taking the second
functional derivatives we will have diagrams (FX1)(FX8) of the set (316)
i a 1 2 1 2 aa 1 1 2 i 
i a b 1 1 i b b 1 1 i a
62 62 4 _
V k k kj j k J b j
(FX1) (FX2) (FX3) (FX4) (FX5) (FX6) (FX7) 2 (FX8) (316)
After the addition of diagrams (V1)(V6) we will have all the diagrams necessary for
building the exchangeonly kernel. During the interpretation of diagrams (V1)(V6) the
(VC7 4) + (VC8 5)
< aclij >< kilab > (< bcA >< kjp > + < bcp >< kjA >) (A62)
a(E E+ Ea EF)(Ei E+ Ea Eb)(EF Ek Ea Eb)
< aclij >< kilab > (< bcA >< kjp > + < bc >< kjA >) (A63)
a,b,c,i,j,k F F F b)
(VC7 6) + (VC8 7)
< aclij >< jklab > (< bcA >< kip > + < bc >< kiA >) (A64)
(A64)
,ci,j,k (i E, E )(Ej + k Ea E,)(cj + k a b)
(VC7 7) + (VC8 4) 
< aclij >< jklab > (< bcA >< kip > + < bc >< kiA >) (A65)
ab,,,j,k ( + aj a )( + j a b)( j + Ek Ea Eb)
a,b,c,i,j,k
(VC7 ) + (VC7 1)
< aclij >< jilad > (< bcA >< bdp > + < bdA >< bcp >)66)
(A66)
,b,,j Ea Ec)( ( + Ea Eb)(Ei E+ a Ed)
(VC7 13) + (VC7 14)
< aclij >< jilab >< bkA >< ckp >
(Ec, i + j a Ec)(i + j Fa Fb)(Ek Ec)
a,b,c,i,j,k
(VC8 2) + (VC8 3)
< ablik >< ljdba > (< jkA >< ilp > + < ilA >< jki >)
(A 68)
,,,j,k, (i + k Ea Eb)(i + j a Eb)(EF +E Ea sb)
(VC8 1) + (VC8 2)
< ablik >< ijlba > (< klA >< jli > + < jlA >< kl/ >)
S+ j ) ( + )( + k (A 6))
a,b,,,i, ja )a
(VC3 9) + (VC 10)
2 < ablik >< ijlab > (< jlA >< Ik > + < jl >< IkA >) (A30)
b,i,k, (i + k EF  b)(Ei +l Fa  b)(Ei + j Ea  b)
(VC3 11) + (VC4 12)
2 < ablik >< ijlac > (< jkA >< bcl > + < bcA >< jkp >) (A31)
,b,ci,j,k (C k Ea Eb)(CF C a Ec)(Eci + c Fa Eb)
(VC3 12) + (VC4 11)
2 < ablik >< ijlac > (< jkA >< bc > + < bcA >< jk >) (A32)
(E,b,c,,,k i + Ek Ea Eb)(Ei + k a Ec)(Ei + Ea Ec)
a,b,c,i,j,k
(VC3 13) + (VC3 14)
< ablik >< ijlab >< jcA >< kcp >
a, + Ek Ea Eb)(i E+ Ea Eb)(Ek Ec)
a,b,c,i,j,k
(VC4 4) + (VC4 6)
< aciij >< ijtdb > (< cbA >< adp > + < adA >< cbi >) (A34)
ab,c,d,i,j + Ea Ec)(i Ea Eb)(Ei Ed Eb)
(VC4 9) + (VC4 10)
2 < aclij >< ijad > (< cb >< bd > + < cb >< bd >) (A35)
,,c,,, (E + Fa a c)(E + Fa Fb)( + a b ( a d)
a,b,c,d,i, j \ c b\z a )
(VC4 13) + (VC4 14)
(A 36)
4 < ac ij >< ijjab >< bkA >< kci > (A36)
S (i e+ C a c)(Ei + j a Eb)(E c)
a,b,c,i,j,k
(VC5 1) + (VC6 1)
2 << ablij >< cjlbk > (< icA >< ak> > + < icl >< akA >)
2 (+ )( ) (A)37)
Our preference in the development of such orbital dependent functionals and
potentials is to start from the density condition that is fundamental to KS theory, as the
single determinant must provide the correct density for the exchangecorrelation problem.
Once this is enforced for a given functional, Vxc is uniquely determined up to a constant.
This is the origin of the perturbation method of Goerling and Levy[14], but essentially
modified by Bartlett et al [16, 17] to avoid the failures of a simple sum of KS oneparticle
Hamiltonians as the unperturbed problem, to generate correlation potentials. This is the
cornerstone of ab initio dft. The conceptual difference is that the density condition does
not explicitly use the variational determination, 6Exc/6p(1) = Vxc(1). This difference is
of critical importance in generating correlation potentials from loworders of perturbation
theory. Without that change, no loworder orbital dependent correlation functional like
that from MBPT2 will generally work, but with those changes, it does very well [16, 48].
Those modifications to the perturbation theory also pertain to the exchangeonly case, the
subject of this paper, but for that problem the distinctions are less important[16].
Once following this approach, which starts with the KS choice of Ho, the distinctions
between applying the density condition and using direct functional differentiation is
more conceptual than essential, as there is a correspondence in any order of perturbation
theory[24, 37].
The direct optimization procedure advocated by some, built upon the above varia
tional determination of the functional derivative, though formally equivalent, differs from
the original, Xbased OEP in the details of implementation. However, as was pointed
out by Staroverov et. al [45] under certain combinations of molecular and auxiliary basis
sets, particularly when the latter's dimension is larger than that for the former, the direct
optimization method can give the HartreeFock energy and density. The latter can be
viewed as a trivial solution, as it can be shown to correspond to the solution of a weighted
leastsquares expression. On the other hand, the Xbased OEP with proper handling is
2
(VC18) (V
(VC18) (VC19)
(VC110)
(VC111) (VC112) (VC113)
(VC114) (VC115)
(VC116)
(320)
Differentiation of diagram (VC2) produce diagrams (VC21)(VC216) of set (321). These
diagrams have the same skeleton structure, but different positions of indices.
k ba ja a
i(VC2) (C22) (C2
(VC211 (VC221 (VC23)
k 01 0 0 1
k ba i b
i( i 6 ( c 1 2 C
2 V 2 2
(VC24) (VC25) (VC26) (VC27) (VC28)
a(C b ja(V b
(VC29) (VC210)
(VC211) (VC212) (VC213) (VC214) (VC215)
(VC216) (321)
After taking the functional derivatives from diagram (VC3) diagrams (VC31) (VC314)
(322) will appear.
2 2
2 a a bj a
2 i j i
a k 1b1 i1 a
k tik kk
i a ia
b 6 b 1 ) b\ 2a
k k a k
(VC31) (VC32) (VC33) (VC34) (VC35) (VC36) (VC37)
J J J J
1 a a 1 na a 5 a
2 ia i' b b Jb b t b b/ b
k i k i k i V k i k iV k k 2
(VC38) (VC39) (VC310) (VC311) (VC312) (VC313)
2 k
G1c k
cib k
b
&k c
(VC314) (322)
TDHF ones. For both excitation energies and polarizabilities this closeness arises from the
fact that the OEP potential derived from the HF energy functional and its timedependent
response properties are similar. Systematic overestimation of polarizabilities is also caused
by the lack of the 1 ..imptotic dependence. In most cases this causes an even larger
error than that coming from the absence of correlation effects in TDHF and TDOEP. The
anisotropy of the polarizability, though, is usually reproduced much better by SVWN.
This fact must be due to cancellation of errors, coming from the incorrect .i,.iinl. l ic and
incomplete cancellation of selfinteraction error, plus a proper accounting of correlation
effects.
5.2.3 ChargeTransfer Excited States
Within the framework of the standard TDDFT description of excited states the
transfer of charge between two separated fragments has severe difficulties. Detailed
analysis of this problem has been presented by Tozer[63] and Dreuw et al[51]. According
to the latter analysis [51] even if the exact exchangecorrelation functional were known, it
would still be impossible to get a proper description of chargetransfer excitations within
TDDFT. Consider the example of He ... Be. The chargetransfer 'I state calculated with
TDHF and TDOEP, taken with the TammDancoff (monoexcited CI) approximation for
simplicity. For both atoms the uncontracted ROOSADZP basis set was used. Potential
curves of the 'I chargetransfer excited state are presented on Fig 52. The potential
curve, calculated with the configuration interaction singles method exhibits the correct
longrange ..imptotic behavior, while the TDOEP results have a qualitatively wrong
.i',i!1 ill. ic behavior. This phenomena is caused by the fact that for chargetransfer
excitations all integrals in Eqn. (536) are equal to zero and the excitation frequency is
equal to the difference of the LUMO orbital energy of the Be atom and the HOMO energy
of the He atom.
What was not addressed numerically in Dreuw et al[51] was the effect of having
a proper selfinteraction and .i', ii, ll ic behavior in DFT, which can only be achieved
potential vertex should be the matrix element < pp\ Vlpq > and a factor 2 must be added
according to equation (310). Interpretation of diagrams (FX1)(FX8) and (V1)(V6)
gives us the expression (317)
h(r3, 6 2
i,j,a,b
< ialjb > (i(3)(ao(r3)(j (r6)L(b(r6)
(Ei Ea)(E. Eb)
2 > l< ij ba > i(r3) oa(r3) j (r6) b(r6)
2
ij, a, F)(E b)
i,j,a,b
+2 y < jkka > [pj(r3) i(r3) i(r6) a(r6) + j (r6)(P(6)(i(r3) o(r3)]
ij,k,a ( Fa)( Fa)
2 y K aij *I > [ao(r3)b(r3)Eb(r6)aj(6) + ( Pa(r6)b(rF6)(cb(r3)i0(3)1
i,j,a,b ( i a
< iklkj > (PF3)La(r3)3j (r6)ao(r6)
,j,k,a ( )( a)
2 y < aj jb > Pi(F3)a(r3) (i 6)b(r6)
(Ei a)(Ei b)
i, j, a,b
2 C < iy ll > i((3 a 3 j (6 + ( ( 6
S< aV lb > y4(r3) a(r3)Pi(r6)ob(r6)
(Ei Ea)(i b)
S [p(r3)fcj(r3)cj (r6)foa(r6) + 6 i6)6j(r 6)(Oj (r3)yOa(F3)1
< i\ [Va > [o(r3)b(r3) b(Tr6)(6T) + Yoa(r6)Yb(r6) b b(r3)3i(r3)1
(E Ea)( Eb)
a,b,i
(317)
Expression (317) for h(r3, r6) exactly corresponds to the expression, obtained by Hirata
et. al, but the diagrammatic derivation requires far less effort and is unambiguous in
terms of signs and numerical factors. Since all the diagrams contain only one contour, it
is possible to make summations in equation (317) only over spatial orbitals, and, as a
result f,,(rl, r2) and faa(rl, r2) will appear separately. The exchangeonly kernel does
not contain the f,3(rl, r2) part, which is a critical difference between the exchange and
correlation kernels.
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
AB INITIO DENSITY FUNCTIONAL THEORY FOR OPEN SHELL SYSTEMS,
EXCITED STATES AND RESPONSE PROPERTIES
By
Denis Bokhan
August 2007
('C! i: Rodney J. Bartlett
Major: Chemistry
Ab initio density functional theory (DFT) based on the optimized effective potential
(OEP) method is a new approach to study the electronic structure of atomic, molecular
and solid state systems. It contains elements of both wave function and density functional
theories and is free from limitations of conventional DFT because of using orbital
dependent functionals derived from systematic approximations of the wave function
theory.
Ab initio DFT methods with exchangecorrelation functionals based on many 
body perturbation theory (MI IPT) have been derived and implemented recently. The
exchangecorrelation potentials derived from MBPT have a complicated structure and
their derivation in higherorder of MBPT by the use of the chainrule for functional
differentiation requires significant effort. To facilitate such derivations, I developed a
special diagrammatic formalism for taking functional derivatives. An alternative way to
construct OEP MBPT exchangecorrelation potentials is to use the density condition.
It makes it possible to obtain potentials for different partitionings of the full molecular
Hamiltonian with MBPT. Using the diagrammatic formalism developed for taking
functional derivatives, we show an orderbyorder equivalence between the functional
derivative and the density condition approaches to OEP MBPT for the case of the Kohn
Sham partitioning of the molecular Hamiltonian. For any other partitionings, different
condition to the density condition that would naturally lead to the analog of the kernel
is necessary to correctly use the OEP2(sc) method. Of course, in infinite order methods,
as opposed to perturbative approximations, there can be no difference between the two
approaches.
3.5 Conclusions
The TDDFT kernel for an OEP method that includes a secondorder MBPT(2)
orbitaldependent correlation functional is derived. This derivation is made possible by a
diagrammatic technique familiar from manybody theory. The complexity of the kernel
for MBPT(2) is excessive, resulting in more than 200 diagrams that have to be evaluated.
This is the price paid to retain the oneparticle structure of DFT and TDDFT yet
introduce rigorous, orbitaldependent correlation functionals even at second order.
Yet, despite the kernel's complexity it has been evaluated in a large basis set and
illustrated for Ne. As only the KS choice of Ho leads naturally to the kernel as the second
functional derivative of the orbitaldependent functional, ELc, the numerical result can
suffer from the fact that the functional E(0) + E(1) + E(2) is not bounded, and without
modification of Ho as used in [16, 17, 26], the radius of convergence is poor, causing poor
convergence[16] in determining the correlation potential. Nevertheless, the first step in
the underlying framework has now been defined to apply TDDFT with OEP correlation
potentials.
However, the complexity is great, though imposition of additional rigorous conditions
might result in simplification of the kernel. Lacking such simplifications, the question
arises as whether this is a case of diminishing returns to evaluate some 203 diagrams to
retain the attractive, oneparticle, correlated structure of DFT? First, the answers for Ne
are not very good at the OEPMBPT(2) level. Furthermore, the basis set dependence
of OEP methods when done in gaussian basis is severe to even get the V,, right[16].
The failure of most such OEP calculations to satisfy the exact HOMO condition <
HOMOIKIHOMO > < HOMOIVHOMO > is a case in point[16, 26].
Hamiltonian
EHF = min < THFIHIHF > (14)
subject to the condition of orthonormality on the orbitals < ,; i; >= 6ij. Substituting
the explicit form of the wave function from Equation (13) into an expectation value, it is
possible to write the HartreeFock energy in terms of orbitals
elec Nucl elec
EHF i 2 ZA j > 1 >)
A RAI
(15)
To derive the HartreeFock equations it is convenient to minimize the following
functional
I = EHF : ij < > (16)
i,j
where Eij are Lagrange multipliers. Minimization of the functional from Equation (16)
requires that the functional derivatives of I with respect to the orbitals vanish
61 61
S0, 0 (17)
6oi 6(p,
Taking into account Equation (13) condition (17) can be presented as
fP=pi, F(18)
The operator f has the structure
f V2 + =Vt+ VH + nl (19)
Let us first consider the choice of Ho as a sum of KohnSham orbital energies:
K
Ho0 = ,{(aa,} (22)
p=1
where K is the number of orbitals.The brackets means that the product of second
quantization operators are written in normal order. In the functional derivative method we
define the exchangecorrelation potential of the nth order as:
V (') (r) = (23)
v 6p(r)
where E(') is the energy of the nth order of manybody perturbation theory. In single
reference manybody perturbation theory we have the energy of the nth order as an
explicit functional of the orbitals.In the G6rlingLevy approach this functional is consid
ered to be a functional for the orbital energies also [14] :
E() E()(v c..... p, 1, .....^ )
Another possible consideration[13],when orbital energies are considered to be a functionals
over corresponding orbitals
E ) = E (')(o .....o C ol [ ]..... ,c[.k ]) (24)
is equivalent to the G6rlingLevy[14] approach.
To construct the chain rule we should take into account the fact that the KohnSham
orbitals are functionals of V,:
E(') = E(')(,(oV,)..... n(V)) (25)
Then the KohnSham potential can be considered to be a functional of p(r):
E() = E (')[I 1(V(p(r))) ..... ,n(V8(p(r)))] (26)
v'(r,t) are alv, different, if
v(r,t) v'(r, t) + c(t) (147)
After invertibility for the G mapping is established, the action functional can be
introduced
A[p] = dt < W[p](t)i HIW[p](t) > (148)
and the variational condition can be applied,
A[p] 0 (149)
6P(r,t)
Using the same manipulations as for the ground state a set of oneparticle equations can
be derived
a (1
p(r, t) ( V2 + ,[p(r, t)])i(r,t) (150)
at 2
where
vs [p(r, t)] =vt(r,t) + Jdr' p(r +. [,1 (r,t) (151)
f r ( )
The great advantage of the timedependent KohnSham scheme lies in its compu
tational simplicity compared to other electroncorrelation methods. The timedependent
KohnSham scheme with explicit timedependence of the density and the potential can
be applied for any type of external, timedependent potentials. However, when the time
dependent external potential is small it can be treated with timedependent perturbation
theory. If the applied perturbation is a periodic electromagnetic field, it is more convenient
to use response theory.
1.2.2 TimeDependent Density Functional Theory Linear Response Theory
Consider the Nelectron system being initially, i. e. at t < to in its ground state. In
this case the initial density po can be calculated from the ordinary groundstate Kohn
Sham equation
(1 2 + vo(r) d' 0 0(plr)) 0)) +0) ) (r) (152)
( 2  r' I r/ ' 0 O i i )(
The perturbation operator is usually defined as the difference between the full and the
zeroorder Hamiltonians
elec elec elec
V = H Ho = t(r) ) + (120)
i i 'i
Introducing A as the small perturbation parameter, the Hamiltonian and the wave
function can be written as
H = Ho + XV (121)
S= + A(1) + A2(2) + ... (122)
E = Eo + + A2AEE + 2E2 ... (123)
These orderbyorder corrections can be found by substituting expansions (121),
(122) and (123) into (116) and collecting terms with the corresponding order in A
(Eo H fo)() >= (V E ())) > (124)
(Eo Ho)(2) >= (V E(1))(1) > E(2) > (125)
C'!....ig the perturbative corrections to be orthogonal to the reference determinant
< T(") I) > 0 (so called intermediate normalization) it is easy to get the expressions for
perturbative corrections at any order. Projecting the equations (124) and (125) onto the
reference determinant expressions for energy corrections are obtained
E(1) =< IV > (126)
E(2) =< IVI(1) > (127)
Expressions for the orderbyorder expansion of wave functions can be written using
the resolvent operator[4]
I(1) >= RoVI) > (128)
(2) > 0(V E(1)) () > Ro(V E 1))RoVI) > (129)
where the c.c means the complexconjugate. Inserting equation (33) into (32) we obtain
: S Jdrz4 (J
p,q p r,sr
d 6E(")
dr.5 s (r5) x
4(,(r5)
x dr6 r X (r6, r6)+ C.)(r4) x
Sr Es
x dr3 Op(r3 )q 3X (r2,3) + C.
Sp Eq
This may be rewritten as (35)
SJdr3 6 (J dr5 ) (P)(r5) X
p,q4p (r3) Ep(r5)
xJdr6 p (r6)q (r6)X (rl,r6) + .c)Xl(r2, r3) + c.c
J p c
Equation (35) can be made more explicit,
P J 3X (r2,r3
p,q p 6V,(r3
dr\
x dr6 P(6)q(76)]X (ri, r6) +c.c) + > J
p p,qp
x f E dr )q (r5)
j JOp(r5)
6E(")
5 q(r5) x
3 p(r5)
dr3X I(r2, r3) X
d p(r6) q (6) X1(r, r6)
dF6 + c(r)
Ep Eq 6Vs(r3)
Using the fact that [14]
6Xl(rl,r2)
6V(r3)
dr drX rr5) X(5, r6)X1(r6,2)
Jdr 6Vs(r3)
it is possible to rewrite equation (36) in the following form (38)
dr3 dr6X(ri, r3)h(3,6)X(r6,2)
Taking into account equation(33) we have the explicit expression (39) for h(r3, r6)
h(r3, r6) V(r3)Y J
p,q p
Edr() p (r6)q (r6)
d (r5 ) r5) q
6()p (r5) Ep Eq
V) ((r5)6X(r5, r6)
dr5 0 6V(rs)
+ c.c
(34)
(35)
(36)
(3 7)
(3 8)
f ()( r2)
f/(n 1, r2)
/f()(rl, 2)
f(P)(r1,r2)
(39)
(VC13 3) + (VC13 4)
2 y fa, < bihaj > (< cjA >< bcp > + < bcA >< cjp >) (A110)
(Ei ,a)(Ej Eb)(Ej Ec)
a,b,c,i,j
(VC13 5) + (VC14 9)
2 fai < kilaj > (< bjA >< bkp > + < bkA >< bjp >) (A
i,j,k,a,b (E a)(j b)(k b)
(VC13 6) + (VC14 6)
2 fai < bilac > (< cjA >< bjp > + < bjA >< cjp >) (A 112)
(Ei Ea)(E'j Eb) (E Ec)
a,b,c,i,j ( )()(
(VC13 13) + (VC15 14) =
4 z < aclik >< bilaj > (< bjA >< ckp > + < ckA >< bjp >) (A
a,,(i E,)(Ej Eb)(Ek Ec)
a,b,c,i,j,k
(VC13 14)
S < aklci >< bilaj >< bjA >< ckp > (A )
2 (A114)
ab,c,i,j,k (Ei a)( Eb)(E Ec)
(VC13 15) + (VC14 14)
2 < aclki >< bilaj > (< bjA >< ckp > + < ckA >< bjp >(115)
2 ~  i  i  ^ (A 1 1 5 )
ab,c,i,j,k (C E,)(Ej b)(Ek c)
(VC14 7) + (VC14 8) =
2 faj < ablij > (< bkA >< ikp > + < ikA >< bkp >) (A 6)
i,j,,,b ( a) a(i b)(Ek b)
(VC14 11) + (VC14 12) =
2 faj < ablij > (< icA >< bcp > + < bcA >< icp >) (A117)
b,c,i,j (Ej Ea) (Ei Eb) (Ei Ec)
(VC10 10) + (VC13 9)
2 fib < iclja > (< abA >< cjp > + < cjA >< abi >)
2 (A86)
(Ei E,)(E Eb)(Ej ,c)
i,j,a,b,c
(VC11 1) + (VC11 2) 
2 z fijfaj(< akA >< iki > + < ikA >< aki >) (A 87)
,j,k,a (i a)(j a)(k a)
(VC11 3) + (VC11 4)
2 z fifaj(< abA >< ib. > + < ibA >< ab. >) (A88)
S(Ej a) (Ei Ea) (Ei Eb)
i,j,a,b b a)
(VC11 6) =
2 fjkfki(< ajA >< aip > + < aiA >< ajp >) (A89)
,j,, (E Fa)(k ,a)(Ej Ea)
(VC11 7) + (VC12 8)
2 fijfab(< bjA >< aip> + < aiA >< bjp >) (A90)
i,j,a,b (i F)(Ei b)(E Eb)
(VC11 9) + (VC12 9)
2 faifjb(< ajA >< bip > + < biA >< ajp >) (A 91)
,j,a,b (i Fa)(j a)( b)
(VC11 o10)
2 faifib < ajA >< bjp > (A92)
(i, i E)( )(j E)(Ej Eb)
i,j,a,b
(VC11 11) + (VC15 5) =
4 fi < ibljk > (< ajA >< kbp > + < kbA >< ajp >) (93)
,j,k,a,b (i a)(j E)(E Eb)
where
6Ec
Vp(r)
j r r/1
To address secondorder properties requires additional effects of the density change given
as the second and higher terms of Taylor's expansion of Ex,. The kernel
f(ri, r2) 
Jp(r2)
is used in [21, 50].However, once our objective is thirdorder molecular property, the
essential new element in TDDFT is the second exchangecorrelation kernel, which in an
adiabatic approximation is defined as the second functional derivative of the exchange
correlation potential with respect to the density
JV1,(ri)
g(ri, 2, r3) =p (61)
6P(r~)6p(r3)
Hence, in this paper we will derive the second kernel for the exactexchange DFT, based
on the OEPx approach. This is a complicated quantity. The traditional way of deriving
kernels and potentials is to tediously derive all terms with the use of the chainrule for
functional differentiation. Yet, even the exchange only kernel, f, has a complicated
structure in OEP, and its further extension to a second kernel is very difficult. To avoid
the use of traditional methods, an effective diagrammatic formalism for taking functional
derivatives was recently developed[37, 62]. In this paper we use that formalism and
apply it to the exchange second kernel for OEPx based timedependent DFT. After
discussing some properties of the second kernel, we report numerical results to obtain
hyperpolarizabilities, compared to those from HartreeFock and coupled cluster singles and
doubles (CCSD).
There is a long history of the treatment of hyperpolarizabilities and associated non
linear optical (NLO) properties. See [69] for a review. In particular, issues of the incorrect
3.3 Properties of the Correlation Kernel
The secondorder correlation kernel has different spin components, which can be
represented by the formula
fn(ri, r2) dr3 dr6Xaa(rl,r3)haar3, r6)Xa (r6,r2)
/ (n r2) f dr df6Xa(ri, r3) 3, 6)X (r6, r2) (3 35)
Diagrams, which contain only one contour cannot produce ha3 components after differenti
ation. Three of the diagrams (VC5)(VC8) after differentiation can produce only diagrams
contributing to the haa or h3p parts. The diagrams (VC91)(VC97), (VC99), (VC910),
(VC101)(VC107), (VC109), (VC1010), (VC111)(VC1110), (VC1113)(VC1116),
(VC121)(VC1210), (VC1213)(VC1216), (VC131)(VC1312), (VC1314), (VC1315),
(VC141)(VC1412), (VC1414), (VC1415) have the same property. Diagrams containing
2 or more contours can produce all spin components. Sets of diagrams (VC17)(VC116),
(VC27)(VC216), (VC37)(VC314), (VC47)(VC414), (VC157)(VC1512) have two
contours, but both 6functions present in one of the contours means that these diagrams
make a contribution to haa or hp parts, but have an extra factor of 2. That factor
appears after summation over all spinorbitals of the second contour. Diagrams (VC1
1)(VC16), (VC21)(VC26), (VC31)(VC36), (VC41)(VC46), (VC98), (VC108),
(VC1111), (VC1112), (VC1211), (VC1212), (VC1313), (VC1413), (VC151)(VC156),
(VC154)(VC1515) have 6 functions on different contours, so they contribute to both haa
and h3p parts. Diagram (VC1513) have an additional factor of 2 and also contribute into
both spin parts. To build all spin parts of the correlation kernel we need to substitute the
above diagrams into equation (335)
The next essential property of the exchange and correlation kernels is the symmetry
with respect to permutation of its arguments
S(r1, 2) f(r2, r1) (336)
potentials are derived from orbitaldependent energy functionals, taken from wave function
theory.
The main advantage of wavefunction based methods is the possibility to obtain
systematically improvable results. This means that with the extension of the basis set, it
will be possible to obtain more accurate energies and wave functions, and in the complete
set it is possible to get the exact solution of the Schr6dinger equation. On the other
hand, rigorous ab initio methods using wave function methods are usually very costly for
computations on large molecules. Ab initio wave function methods can usually be applied
only for systems with ~2030 atoms in reasonable basis sets. The main DFT advantage
is a small computational cost; it can handle systems with several hundreds of atoms.
However, most of conventional density functionals do not have the capacity to produce
systematically improvable results.
Ab initio DFT is capable of producing systematically improvable results, but it
is computationally more costly than DFT with conventional functionals. Despite its
computational cost, ab initio DFT can be used as a method for the calibration of density
dependent functionals. An alternative way of providing such information is Quantum
MonteCarlo (QMC) method, however, QMC results are not available for molecules and
openshell systems.
1.1 Ab Initio WavefunctionBased Methods
1.1.1 HartreeFock Method
In the HartreeFock method, a wave function is considered as a Slater determinant[1]
0 (ri) ...* ci(rN)
HF= : (13)
(PN(r1) ... (PN(rN)
The singleelectron wave functions 4p(r) (or orbitals) are determined by the condition
that the corresponding determinant minimizes the expectation value of the electronic
LIST OF TABLES
Table page
31 Orbital energies and zeroorder approximations to excitation energies . 64
32 Excitation energies of Ne atom using OEPMBPT(2) KohnSham orbital energies 64
33 Excitation energies of Ne atom using exchangeonly orbital energies ...... ..66
34 Excitation energies of Ne atom using orbital energies and orbitals from OEP2(sc).
All equations for TDDFT are the same .................. ...... 66
41 Total energies .................. ................. .. 72
42 Ionization potentials (in e. v.) .................. ........ .. 74
43 Dissociation energies (in kJ/mol) .................. ....... .. 75
44 Singlet and triplet energies of methylene .................. ..... 76
51 Total (in a. u.) and orbital (in e. v.) energies of Ne atom . ..... 88
52 Total and orbital energies of He atom .................. ...... 88
53 Excitation energies (V valence state, R Rydberg) .............. ..93
54 Orbital energies (in e. v.) of Ne atom .................. ...... 94
55 Ionization energies (in e. v.) .................. ......... .. 94
56 Static polarizabilities (in a. u.) .................. ........ .. 99
57 Isotropic C6 coefficients (in a. u.) .................. ..... .. 99
61 Hyperpolarizabilities of several molecules (in a. u.) ............... ..107
BIOGRAPHICAL SKETCH
I was born in Avdeevka, a small town located in the southeastern part of the
Ukraine in the Donetsk region. I grew up in the city of Gorlovka, a workingclass mining
community. When I was in ninth grade, I acquired an interest in chemistry and decided
that I would become a chemist. During my last year of high school, I was one of the win
ners of the Ukrainian ('!, iiiI ry Olympiad. As a result, I was admitted to the prestigious
Moscow State University without the otherwise necessary entrance examinations. During
my sophomore year of college, I decided to specialize in quantum chemistry. After I de
fended my undergraduate thesis and obtained my degree in 2002, I began to work toward
joining Dr. Rodney Bartlett's research group at the University of Florida. After working
as a system administrator for several companies in Moscow for 16 months, I obtained the
necessary funds to l i, for transportation to the United States as well as for the necessary
exams, which included the Graduate Record Examination (GRE) and the Test of English
as a Foreign Language (TOEFL). Finally, I joined Dr. Bartlett's group at the University
of Florida on January 21st, 2004.
CHAPTER 6
EXACTEXCHANGE TIMEDEPENDENT DENSITY FUNCTIONAL THEORY FOR
HYPERPOLARIZABILITIES
The timedependent OEP exchangeonly method (OEPx) has been previously
derived by G6rling[20] and implemented for molecules by Hirata et. al [21, 50]. Numerical
results in the latter papers show a reasonable description of both valence and Rydberg
excited states. Reasonable static and dynamic polarizabilities are also obtained. Good
results of TDOEPx for excited states and properties are obtained partly due to the correct
.ivinl Il' '1 ic behavior of the exchange potential, but also due to the elimination of the
selfinteraction error. In particular, OEPTDDFTx tends to be superior to standard
approaches like TDDFT based upon the local density approximation (LDA) or BeckeLee
YangParr (BLYP) functionals. Similarly, exchangeonly OEP with exact local exchange
(EXX) [22, 23] has been shown to greatly improve bandgaps in polymers [23]
Another advantage of OEP based methods is that since virtual orbitals in the
exchangeonly DFT as well as the occupied orbitals are generated by a local potential,
which corresponds to the Nl particle system due to the satisfaction of the selfinteraction
cancellation, the differences between orbital energies of virtual and occupied orbitals
offer a good zerothorder approximation to the excitation energies[16, 17]. This is not
possible in the case of HartreeFock theory, without adding a VN1 potential[67, 68],
since occupied orbitals are generated by an Nl particle potential and the energies of
unoccupied orbitals come from Nelectron potential, and thereby, approximate electron
affinities.
In KohnSham DFT
hKS = T + Vezt + J + V1e
2.3 Interconnection in Arbitrary Order
2.3.1 Diagrammatic Functional Derivatives
For formulation of the rules we will use equation (218). The 6 function needs to be
introduced according to equation (226):
b
aa
6 (rqP(r) + + i+ a (226)
p,qPp q
Brackets on the last two diagrams denote denominators. Expressions for the energy
in MBPT consist of linear combinations of terms, which have products of molecular
integrals in the numerators and product of differences of oneparticle energies in the
denominators, so it is easy to make a diagrammatic representation for such expressions.To
take the functional derivatives from numerators, according to equation (218), all lines
connected to some vertex must be disconnected from the corresponding place of linking,
and when this is done, a new line (corresponding to occupied or unoccupied orbitals) must
be inserted. In the final step contraction with the corresponding 6 function must be
provided. All lines corresponding to denominators are still unchanged. This procedure
must be provided for all vertices,because when we take functional derivatives from
products of functions, we have a sum of products, according to rules for taking derivatives.
If a new diagram changes its sign, a minus sign must be assigned to this diagram.
When functional derivatives from denominators are taken, it is more suitable to use
equation (213) The most general form for the denominator can be represented by the
following formula:
Ka Ka
Den= c( fia V E )) (227)
a i=l j=1
(VC6 5) + (VC8 9)
2 z < ablij >< likb > (< akA >< jl/ > + < jlA >< akp >)54)
,b,, (E + j j a Eb)( + Fl Fa Eb)(k Ea)
(VC6 7)
2 < allij >< jilkb >< akA >< lbp > (A55)
a,b,i,j,k, + E Ea b) (E a)(E b)
(VC6 9)
2,, < lblij >< ak >< lap >56)
,,k, (Ei + c Ea b)(Ek a E)(El Ea)
(VC6 10) + (VC7 9)=
2 < ablij >< jilkb > (< ckA >< acp > + < ck >< acA >) (A57)
E,, (+ j   Eb)(Ei + E,  Eb)(Ek Ec)
a,b,c,i,j,k
(VC6 11) + (VC6 2)
2 < cb ij >< jilkb > (< akA >< acp > + < akp >< acA >) (A
2 +b)( k k) (A58)
alb,c,i,j,k (Ei E+ Ec Eb)(EFk Ec)(Ek Ea)
(VC6 13) + (VC6 14)
2 < ablij >< jikb > (< klA >< al > + < a >< kl >) (A59)
a,b,i,j,k,l ( E+ F Eb)(l a)(Ek a)
(VC6 16) =
< ablij >< jilcb >< akA >< kcp >
2 (A60)
,,,. (Ei Fa Eb)(Ek a)(Ek c)
a,b,c,i,j,k
(VC7 2) + (VC7 3)
< aclij >< jildb > (< bcA >< adp > + < adA >< bcp >)
,c,d,i,j (Ei + F Ec)(Ei + j a Eb)(Ei j bd Eb)
a,b,c,d,i,j
Consider the second and third diagram in the energy expression. For the sake of simplicity
only nonequivalent diagrams will be given. Taking functional derivatives with respect to
occupied orbitals we will have equation (231)
i J bi + (231)
Inserting the lines of unoccupied orbitals and making contractions with the 6 function,
diagrams 4 and 8 of equation (225) will be obtained (232):
C C
a a
2 4 +2 8 (232)
When functional derivatives are taken with respect to unoccupied orbitals and occupied
orbitals are inserted, diagrams 5 and 9 of equation (225) will be obtained.When this
procedure is done for the lower vertex, complex conjugate diagrams will be obtained
and we will have the same number of diagrams of this sort as in the effective operator
approach.
When functional derivatives are taken with respect to occupied orbitals for the
upper and lower vertexes and an occupied line is inserted, we have the diagrams, given by
equation (2 33)
k a 1i a k a
a b b jjk a b j k a b j
222 2 2 (233)
"Brackettype" denominator means the difference Ei Ek. Summation of the fist two
diagrams according to the FrantzMills theorem[18] and the same procedure for the second
CHAPTER 1
INTRODUCTION
The definition of a chemical reaction as a transformation of one substance to another
at the molecular level means a regrouping of nuclei and electrons. In the adiabatic
approximation, such nuclear regroupings can be presented as a motion in the field of
some potential, also known as a potential energy , if',.. Potential energy surfaces can be
obtained by solving the eigenvalue problem with the so called electronic Hamiltonian
1 lec elec Nucl ZA elec 1
2 E Iri RA I ri rj
i=1 i=1 A=1 i
Whole nuclear coordinates are treated as a set of parameters. If there are more than two
particles in the system, the eigenvalue problem
H,i(R...RM)(rl... rN, R1...RM) = E(R ...RM)1(rl... rN, R... RM) (12)
can not be solved analytically. For the calculation of electronic energy levels, some
approximations have to be used.
Modern quantum chemistry includes three classes of methods for the solution
of the electronic problem. The first one, called wave function 'I' ,., '; uses different
approximations for the calculation of the wave function '(rl...rN, R1...RM) and the
corresponding energy E(R1...RM). The second class, known as /. ,:.:/;/ functional theory
uses the density as a primary object. Within the DFT approach, the energy is written as
a functional of density, and the construction of the wave functions is not necessary. The
third class of methods uses a whole density matrix. The density matrix renolmalization
group (DMRG) and the density matrix functinal theory (DMFT) are typical methods of
the third class.
Ab initio DFT contains elements of the first two classes. It uses a local multiplicative
potential, typical for the DFT approach. However, the corresponding exchangecorrelation
3.2 Kernel for the SecondOrder Optimized Effective Potential ManyBody
Perturbation Theory Correlation Potential
For the derivation of the secondorder exchangecorrelation kernel we require
functional derivatives of the secondorder correlation energy with respect to V,. These
functional derivatives can be represented by diagrams (VC1)(VC15) of (318).
c ^ k
a b j b j6 a
(VC1) (VC2) (VC3) (VC4)
b k
ba
c b 1 i\ /a b
\7 \/
(VC5) (VC6) (VC7) (VC8)
b i F
CF
j c
F a F a b
a a
a a i a i b j
F FF F
(VC9) (VC10) (VC11) (VC12)(vC13) (VC14) (VC15) (318)
Diagrams (VC1), (VC2), (VC5), (VC6) and (VC15) of (318) have an external factor 4,
diagrams (VC9) and (VC10) have no factor, while the rest of the diagrams have a factor
of 2. To get the correlation kernel diagrams we need to take functional derivatives with
respect to the potential from diagrams (V1)(V15). After applying the diagrammatic
rules to diagram (VC1) and the use of relation (313), diagrams (VC11)(VC116) of sets
(319) and (320) will appear.
~l _2 /01 2 f1 k1 C1 C1 A0,
c k cd ci
b i Ji j i b b i i b j
S
(VC11) (VC12) (VC13) (VC14) (VC15) (VC16) (VC17) (319)
behavior of TDDFT hyperpolarizabilities for NLO design for polymers[64, 65] points to
the necessity of an improved theory. A rigorous (ab initio) DFT analogue starts with
exchangeonly, TDOEPx.
6.1 Theory
6.1.1 TimeDependent Density Functional Theory Response Properties
When an external timedependent electric field is applied, the perturbation can be
written in the form (62)
V(t) = Z r E(t) (62)
For the case of periodic perturbations, the virtualoccupied block of the linear response of
the density can be found from the following system of linear equations
(A + B)U = h (63)
The matrices A, B have the structure, given by Eqn (536) and h is
haa =< a, r i, > (64)
The calculation of the components of the first, static hyperpolarizability tensor
requires [70]
S=2 (U F U UF U +UF U +
i,p,q,(
!3A~w 2Z(U> F qJ zpU(;Jqa pq?,qia i~pa pqcT qicI
UFP U FU U UP, F, U) +
2 (U\. c UJ + UX U + U +
i,j,p,u
U + U" U U U ) +
8S< baubc gj i Pib\ j9T\ k > lUa Utjvk (65)
a ab,c,i,j,k
(VC1 4) + (VC4 1)
< adlij >< cjab > (< icA >< db/ > + < ic/ >< dbA >)
(i + Ej Ea Eb)(i + Ej a Ed)(Ei Ec)
(VC1 5)=
2 < ablid >< cj ab >< icA >< dji >
a,b,c,d,i,j (i + j b)(i c)(j d)
(VC1 6) + (VC2 5)
< ak ij >< cjab > (< icA >< kbp > + < kbA >< icp >)
S(i + Ej a Eb)(Ei Ec)( k Eb)
(VC1 7)
< abldj >< cjlab >< icA >< idi >
a,b,c, Ea Eb)(Ei Ec)(Ei Ed)
a,b,c,d,i,j
< kblij >< cjlab >
(Ei + Ej
< abiij >< cjab >
(Ei + Ej
< abiij >< cjab >
(Ei + Ej a
(VC1 8) + (VC2 8)
(< icA >< ka/ > + < kaA >< icp >)
 Eb)(Ei Ec)(k Ea)
(VC1 9) + (VC1 10)
(< ikA >< kci > + < kcA >< iki >)
 Eb)(i ( c)(k c)
(VC 11) + (VC3 7)
(< ikA >< kci > + < kcA >< iki >)
 Eb)(Ek + Ej a Eb)(Ek Ec)
(VC1 12) + (VC4 8)
< ablij >< cjldb > (< icA >< dap > + < ic/ >< daA >)
(Ei + Ej a b)(Ei + Ej d Eb)(Ei Ec)
(A10)
(All)
(A12)
(A13)
2
ab,c,d,i,j
2 Z
a,b,c,i,j,k
(A6)
(A7)
(A8)
(A9)
4 a
a,b,c,i,j,k
4 E
a,b,c,i,j,k
4 a
a,b,c,i,j,k
4
ab,c,d,i,j
terms of the density, the KohnSham theorem[6] can be used. This theorem states, that
the groundstate energy as a functional of density has a minimum, if the density is exact.
Therefore, given the energy functional, one can obtain the groundstate density and energy
by variational minimization of the functional. The formal definition of DFT does not
tell how to construct such a functional. Several approximate forms have been iri. 1. I
however, whose accuracy varies greatly for different properties. The kinetic energy of
electronic motion is particularly difficult to approximate as a density functional.
The basic idea of Kohn and Sham was to transform the variational search over the
density into a search over the orbitals that integrate to a given trial density
( V2 + v(r))P'(r) = p,(r) (135)
Such a transformation does not restrict the variational space, provided that every phys
ically meaningful density corresponds to a unique set of orbitals (the vrepresentability
condition). The use of the KohnSham SCF model ensures not only that the variational
density is a fermionic density, but it also provides a good approximation for the kinetic
energy. If orbitals integrate to the true density it is natural to expect that
Sdec
T, < V21> (136)
i
accounts for a large part of real kinetic energy.
The rest of the unknown terms are grouped into the exchangecorrelation functional
Ec[p] = E[p] T, Eet EH (137)
where EH is a Hartree energy and can be easily be calculated from a given set of Kohn
Sham orbitals. The noninteracting kinetic energy T, is supposed to reproduce a large
part of the exact kinetic energy T. Therefore, Ec is easier to approximate as a density
functional than the total energy E.
2
a,b,c,d,i,j
(VC5 2)
< ablij >< cdlba >< icA >< djp >
(E + Ej a Eb)(i E,) Ej Ed)
(VC5 3) + (VC8 1)
2 < ablik >< cjba > (< icA >< jk/ > + < jkA>)
ab,c,i,j,k (c( + F i a Eb)(Ei+ Ek Ea Eb)(Ei Ec)
(VC5 4) + (VC7 12)
2 < dbjij >< cjba > (< icA >< dap > + < daA >< icp >)
~,b,cd(i +C d Ib)( + Ea b)(Ei Ec)
a,b,c,d,i,j
(VC5 5) + (VC7 8) =
2 < abiij >< cjda > (< icA >< dbi > + < dbA >< icp >)
,c,d,i, (E Ej Ea b)(E Ej Fa E Ed) (E Ec)
a,b,c,d,i,j
Z < aklij >< cjlba >
a,b,c,i,j,k ( 
2 Z
a,b,c,d,i,j
< ablid
(Ei + Fj
2 < kblij >< cjlba >
b,,i,j,k (Ei + Ej
a,b,c,i,j,k
(VC5 6) + (VC6 8)
(< icA >< kb/ > + < ic/ >< kbA >)
 Ea Eb)(i Ec)(Ek Eb)
(VC5 7)
>< cjbba >< icA >< jbp >
 a Eb)(E Ec)(E ? d)
(VC5 8) + (VC6 6)
(< icA >< kai > + < icp >< kaA >)
 a Eb)(Ei ,c)(k a)
(VC5 9)
>< cba >< ic >< idi >
 E Eb)(i Ec)(Ei d)
(A38)
(A39)
(A40)
(A41)
(A42)
(A43)
(A44)
< abldj
(Ei + Ej
2
a,b,c,d,i,j
(A45)

Full Text 
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IwouldliketothankprofessorsHenkMonkhorstandSoHirataforhelpfulldiscussions.IwanttothankalsoDr.IgorSchweigertandDr.NorbertFlockeforthehelpingmetowriteOEPcode.MyspecialthankstoTatyanaandThomasAlbertforhelpwiththepreparationofthedissertationtext. 3
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page ACKNOWLEDGMENTS ................................. 3 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 12 1.1AbInitioWavefunctionBasedMethods .................... 13 1.1.1HartreeFockMethod .......................... 13 1.1.2ElectronCorrelationMethods ..................... 16 1.2KohnShamDensityFunctionalTheory .................... 19 1.2.1TimeDependentDensityFunctionalTheory ............. 21 1.2.2TimeDependentDensityFunctionalTheoryLinearResponseTheory 23 1.2.3ProblemswithConventionalFunctionals ............... 26 1.2.4OrbitalDependentFunctionals ..................... 28 1.2.5AbInitioDensityFunctionalTheory ................. 29 2INTERCONNECTIONBETWEENFUNCTIONALDERIVATIVEANDEFFECTIVEOPERATORAPPROACHESTOABINITIODENSITYFUNCTIONALTHEORY .................................. 31 2.1EquationsfortheExchangeCorrelationPotentialintheFunctionalDerivativeApproach .................................. 31 2.2EquationsfortheExchangeCorrelationPotentialinanEectiveOperatorApproach .................................... 34 2.3InterconnectioninArbitraryOrder ...................... 37 2.3.1DiagrammaticFunctionalDerivatives ................. 37 2.3.2DiagrammaticFunctionalDerivativesinSecondOrderManyBodyPerturbationTheory .......................... 38 2.3.3InterconnectioninHigherOrders ................... 41 2.3.4InterconnectioninInniteOrder .................... 45 3ABINITIOTIMEDEPENDENTDENSITYFUNCTIONALTHEORYEMPLOYINGSECONDORDERMANYBODYPERTURBATIONOPTIMIZEDEFFECTIVEPOTENTIAL ............................. 47 3.1DiagrammaticConstructionoftheExchangeCorrelationKernels ...... 48 3.1.1Formalism ................................ 48 3.1.2AnExample:DiagrammaticDerivationofExchangeOnlyKernel .. 52 4
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.................. 54 3.3PropertiesoftheCorrelationKernel ...................... 61 3.4NumericalTesting ................................ 62 3.5Conclusions ................................... 67 4ABINITIODENSITYFUNCTIONALTHEORYFORSPINPOLARIZEDSYSTEMS ....................................... 69 4.1Theory ...................................... 69 4.2ResultsandDiscussion ............................. 71 4.2.1TotalEnergies .............................. 71 4.2.2IonizationPotentials .......................... 74 4.2.3DissociationEnergies .......................... 75 4.2.4SingletTripletSeparationinMethylene ................ 75 4.3Conclusions ................................... 75 5EXACTEXCHANGETIMEDEPENDENTDENSITYFUNCTIONALTHEORYFOROPENSHELLSYSTEMS ........................ 80 5.1ExactExchangeDensityFunctionalTheory ................. 82 5.2TimeDependentOptimizedEectivePotential ................ 91 5.2.1TheoryandImplementation ...................... 91 5.2.2NumericalResults ............................ 92 5.2.3ChargeTransferExcitedStates .................... 95 5.3Conclusions ................................... 97 6EXACTEXCHANGETIMEDEPENDENTDENSITYFUNCTIONALTHEORYFORHYPERPOLARIZABILITIES ..................... 100 6.1Theory ...................................... 102 6.1.1TimeDependentDensityFunctionalTheoryResponseProperties .. 102 6.1.2DiagrammaticDerivationoftheSecondExactExchangeKernel .. 103 6.1.3PropertiesoftheSecondExactExchangeKernel ........... 106 6.2NumericalResults ................................ 107 6.3Conclusions ................................... 109 APPENDIX AINTERPRETATIONOFDIAGRAMSOFTHESECONDORDERMANYBODYPERTURBATIONTHEORYOPTIMIZEDEFFECTIVEPOTENTIALCORRELATIONKERNEL ............................. 110 BINTERPRETATIONOFDIAGRAMSOFEXACTEXCHANGESECONDKERNEL ....................................... 127 REFERENCES ....................................... 131 5
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................................ 135 6
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Table page 31Orbitalenergiesandzeroorderapproximationstoexcitationenergies ...... 64 32ExcitationenergiesofNeatomusingOEPMBPT(2)KohnShamorbitalenergies 64 33ExcitationenergiesofNeatomusingexchangeonlyorbitalenergies ....... 66 34ExcitationenergiesofNeatomusingorbitalenergiesandorbitalsfromOEP2(sc).AllequationsforTDDFTarethesame ....................... 66 41Totalenergies .................................... 72 42Ionizationpotentials(ine.v.) ........................... 74 43Dissociationenergies(inkJ/mol) .......................... 75 44Singletandtripletenergiesofmethylene ...................... 76 51Total(ina.u.)andorbital(ine.v.)energiesofNeatom ............ 88 52TotalandorbitalenergiesofHeatom ....................... 88 53Excitationenergies(Vvalencestate,RRydberg) ............... 93 54Orbitalenergies(ine.v.)ofNeatom ....................... 94 55Ionizationenergies(ine.v.) ............................ 94 56Staticpolarizabilities(ina.u.) ........................... 99 57IsotropicC6coecients(ina.u.) ......................... 99 61Hyperpolarizabilitiesofseveralmolecules(ina.u.) ................ 107 7
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Figure page 41ExchangeandcorrelationpotentialsofLiatom(radialpart).A)Exchangepotential.B)Correlationpotential ........................... 73 42ExchangeandcorrelationpotentialsofO2moleculeacrossthemolecularaxis.A)Exchangepotential.B)Correlationpotential .................. 73 43LiHpotentialenergycurve. ............................. 77 44OHpotentialenergycurve. .............................. 78 45HFpotentialenergycurve. .............................. 79 51ExchangepotentialsofNeatom,obtainedindierendbasissets ......... 88 52A1chargetransferedexcitedstateofHe...Be ................. 96 53LUMOHOMOorbitalenergydierence ...................... 96 8
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^Hel=1 2elecXi=1r2ielecXi=1NuclXA=1ZA ^Hel(R1:::RM)(r1:::rN;R1:::RM)=E(R1:::RM)(r1:::rN;R1:::RM)(1{2)cannotbesolvedanalytically.Forthecalculationofelectronicenergylevels,someapproximationshavetobeused.Modernquantumchemistryincludesthreeclassesofmethodsforthesolutionoftheelectronicproblem.Therstone,calledwavefunctiontheory,usesdierentapproximationsforthecalculationofthewavefunction(r1:::rN;R1:::RM)andthecorrespondingenergyE(R1:::RM).Thesecondclass,knownasdensityfunctionaltheoryusesthedensityasaprimaryobject.WithintheDFTapproach,theenergyiswrittenasafunctionalofdensity,andtheconstructionofthewavefunctionsisnotnecessary.Thethirdclassofmethodsusesawholedensitymatrix.Thedensitymatrixrenolmalizationgroup(DMRG)andthedensitymatrixfunctinaltheory(DMFT)aretypicalmethodsofthethirdclass.AbinitioDFTcontainselementsofthersttwoclasses.Itusesalocalmultiplicativepotential,typicalfortheDFTapproach.However,thecorrespondingexchangecorrelation 12
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1.1.1HartreeFockMethodIntheHartreeFockmethod,awavefunctionisconsideredasaSlaterdeterminant[ 1 ] HF='1(r1)'1(rN)......'N(r1)'N(rN)(1{3)Thesingleelectronwavefunctions'i(r)(ororbitals)aredeterminedbytheconditionthatthecorrespondingdeterminantminimizestheexpectationvalueoftheelectronic 13
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1{3 )intoanexpectationvalue,itispossibletowritetheHartreeFockenergyintermsoforbitals 2r2NuclXAZA 2elecXi;j(<'i'jj'i'j><'i'jj'j'i>)(1{5)ToderivetheHartreeFockequationsitisconvenienttominimizethefollowingfunctional 1{6 )requiresthatthefunctionalderivativesofIwithrespecttotheorbitalsvanish 'i=0;I 'i=0(1{7)TakingintoaccountEquation( 1{3 )condition( 1{7 )canbepresentedas 2r2+^vext+^vH+^vnlx(1{9) 14
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^vext(r)=elecXiNuclXAZA (1{10) TheHartreeFockexchangeoperator^vnlxisnonlocal,i.e.itcannotbepresentedasananalyticalfunctionofspatialvariables.However,itispossibletowriteitsactiononsomeorbital'i 1{8 )canberewritteninthecanonicalform 15
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FCI=HF+occXivirtXaCaiai+occXi6=jvirtXa6=bCabijabij+:::(1{14)whereai,abij,etcareSlaterdeterminants,formedbysubstitutionofoccupiedorbitalsi,j...byvirtualorbitalsa,b...withthecorrespondingreorderingofrows.TheexpansioncoecientsarefoundfromthevariationalconditionontheexpectationvalueoftheHamiltonian (1{15) 16
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2 ]andManyBodyPerturbationTheory[ 3 ].AnytruncatedversionoftheCImethodhasaqualitativelywrongbehavioroftheenergiesandwavefunctionswhileincreasingthenumberofparticlesinthesystem.Therefore,theCImethodswithlimitedlevelofexcitationscannotbeusedforhighlycorrelatedsystems.TheCoupledClustermethodandManyBodyPerturbationtheoriesarefreefromthislackofextensivityfailureandareverycommonforthemolecularcomputations.InsomecasesperturbationtheorycanprovideanaccuratedescriptionofelectroncorrelationeectsatamuchlowercostthannecessaryfortheCoupledClustermethod.ThesecondorderRayleighSchrodingerperturbationtheoryisthesimplestandleastexpensiveabinitiomethodfortakingintoaccountelectroncorrelationeects.InthisperturbationtheorythesolutionofSchrodingerequation ^H=E(1{16)canbefoundusingtheSlaterdeterminantasareference.Generally,suchadeterminantmaybeconstructedfromtheorbitals,generatedbysomeoneelectronoperator ^h'p=(1 2r2+^u)'p="p'p(1{17)TherststepofanyperturbationtheoryisthepartitioningoftheHamiltonianintoazeroorderH0andperturbation ^H=^H0+^V(1{18)where ^H0=E0=Xi(hi)=(elecXi"i)(1{19) 17
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^V=^H^H0=elecXi^vext(ri)elecXi^u(ri)+elecXi6=j1 ^H=^H0+^V =+(1)+2(2)+::: Theseorderbyordercorrectionscanbefoundbysubstitutingexpansions( 1{21 ),( 1{22 )and( 1{23 )into( 1{16 )andcollectingtermswiththecorrespondingorderin (E0^H0)j(2)>=(^VE(1))j(1)>E(2)j> Choosingtheperturbativecorrectionstobeorthogonaltothereferencedeterminant<(n)j>=0(socalledintermediatenormalization)itiseasytogettheexpressionsforperturbativecorrectionsatanyorder.Projectingtheequations( 1{24 )and( 1{25 )ontothereferencedeterminantexpressionsforenergycorrectionsareobtained Expressionsfortheorderbyorderexpansionofwavefunctionscanbewrittenusingtheresolventoperator[ 4 ] 18
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^R0=^Q E0^H0(1{30)and^Q=1j>.Since^H0isdiagonalinthebasisofSlaterdeterminants,itispossibletowrite ^R0=Xn6=0jn>() 1{17 ),thesecondordercorrectiontotheenergyhasthefollowingstructure 2occXi;junoccXa;b() 5 ],establishesaonetoonemappingbetweenthegroundstateelectronicdensityandtheexternalpotential.Theexternalpotentialdenesaparticularobject(atom,molecule,etc)and,becauseoftheonetoonemapping,thedensitycontainsalltheinformationaboutthesystem.Inparticular,thegroundstateenergycanbewrittenasafunctionalofthedensity.Togetthegroundstateenergyin 19
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6 ]canbeused.Thistheoremstates,thatthegroundstateenergyasafunctionalofdensityhasaminimum,ifthedensityisexact.Therefore,giventheenergyfunctional,onecanobtainthegroundstatedensityandenergybyvariationalminimizationofthefunctional.TheformaldenitionofDFTdoesnottellhowtoconstructsuchafunctional.Severalapproximateformshavebeensuggested,however,whoseaccuracyvariesgreatlyfordierendproperties.Thekineticenergyofelectronicmotionisparticularlydiculttoapproximateasadensityfunctional.ThebasicideaofKohnandShamwastotransformthevariationalsearchoverthedensityintoasearchovertheorbitalsthatintegratetoagiventrialdensity (1 2r2+vs(r))'p(r)="p'p(r)(1{35)Suchatransformationdoesnotrestrictthevariationalspace,providedthateveryphysicallymeaningfuldensitycorrespondstoauniquesetoforbitals(thevrepresentabilitycondition).TheuseoftheKohnShamSCFmodelensuresnotonlythatthevariationaldensityisafermionicdensity,butitalsoprovidesagoodapproximationforthekineticenergy.Iforbitalsintegratetothetruedensityitisnaturalytoexpectthat 2elecXi<'ijr2j'i>(1{36)accountsforalargepartofrealkineticenergy.Therestoftheunknowntermsaregroupedintotheexchangecorrelationfunctional 20
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2Zdr1dr2(r1)(r2) 21
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7 ].WecanstartfromthetimedependentSchrodingerequation (t0)=0(1{43)undertheinuenceofdierentexternalpotentialsv(r,t).Foreachxedinitialstate0,theformalsolutionoftheSchrodingerequation( 1{42 )denesamap 22
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@t^Hj[](t)>(1{48)andthevariationalconditioncanbeapplied, @t'i(r;t)=(1 2r2+vs[(r;t)])'i(r;t)(1{50)where (1 2r2+v0(r)+Zdr00(r0) 23
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1{50 )provideaformallyexactwayofcalculatingthetimedependentdensity,itispossibletocomputeexactdensityresponse1(r;t)astheresponseofthenoninteractingsystem 24
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1{55 )and( 1{56 )arethebasisoflinearresponsetheory.Sinceequation( 1{55 )isnotlinearwithrespectto1thesolutionshouldbeobtainedbysomeiterativeprocedure.However,forallpracticalpurposesadirectuseofequation( 1{55 )isnotconvenient.Togetamoreconvenientform,thefollowingmatrixelementshouldbeconsidered (1{59) Ontheotherhand,<'aj1(r;!)j'i>canbeexpressedintermsofaresponsefunction !+("a"i);Pai(!)=<'ijv1(!)j'a> 1{61 )intoequation( 1{59 )wehave (1{62) Introducingthenotations 25
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1{62 )canberewritteninthefollowingmatrixform 1{64 )canberewrittenas[ 8 { 11 ] 1{65 )iscommonforthecalculationofexcitationenergieswithtimedependentdensityfunctionaltheory(TDDFT).MostmodernimplementationsofTDDFTusethefrequencyindependentexchangecorrelationkernels,denedasasimplefunctionalderivative 26
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27
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28
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2Xi;j<'i'jj'j'i>(1{67) 29
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30
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12 13 ]isastraightforwardapproachtotheabinitioDFT.ThecorrespondingKohnShampotentialcanthenbeobtainedbytakingthefunctionalderivativeoftheniteorderenergyviathechainrule,thattransformsthederivativewithrespecttothedensityintothederivativewithrespecttoorbitalsandorbitalenergies[ 14 ].ThisleadstotheOptimizedEectivePotential[ 13 15 ]equationswhichcanbecomplicatedalreadyinsecondorder[ 13 ]Alternatively,onecandeterminetherstandsecondorderKohnShampotentialbyrequiringthatthecorrespondingrstandsecondorderperturbativecorrectionstothereferencedensityvanish.Unlikethefunctionalderivativeapproachsuchaconditiononthedensitycanbedescribedwithstandardmanybodytechniques.Therecentwork[ 17 ]usesdiagramstoderivethesecondorderOEPequationinasystematicandcompactfashion,whileasecondpaper[ 16 ]doessoalgebraically.Boththefunctionalderivativeanddensity(eectiveoperator)approachesleadtoexactlythesameequationintherstorder[ 17 ].However,thefunctionalderivativeofthesecondenergyinvolvesacertaintypeofdenominatorsthatarenotpresentinthedensitycondition(eectiveoperatorapproach).Stillthetermsinvolvingsuchdenominatorscanbetransformedtomatchthedensitybasedequationexactly.Thiscaveatraisesthequestionofwhetherthetwoapproachesareequivalentinhigherorders,orfordierentpartitioningsofthehamiltonian. 31
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14 ]:E(n)=E(n)('1:::::'n;1:::::n)Anotherpossibleconsideration[ 13 ],whenorbitalenergiesareconsideredtobeafunctionalsovercorrespondingorbitals 14 ]approach.ToconstructthechainruleweshouldtakeintoaccountthefactthattheKohnShamorbitalsarefunctionalsofVs: 32
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(2{8) (2{9) Usingequations( 2{8 )and( 2{9 )wecanrewritethevariationoftheenergyinthefollowingform: 13 14 ]forthecalculationofV(n)xc: 14 ]: 13 ] 33
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2{15 )as: "i"a=Xp;q6=p "i"a=Xa;ifa+paqg(2{21) 34
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17 ].Thismeans,thatifweconstructaneectiveoperatorofthedensityusingMBPT,thecorrectiontothedensitymustvanishinanyorder.Thisisthemainideaoftheeectiveoperatorapproach.Fortheexchangecorrelationpotentialofrstorderwewillhave: 2{23 ). (1)(2{23)Usingthefactthatfpq="ppq+<'pjVHFxVx()Vcj'q>,wecanextractourdesirableexchangepotential.Thecorrelationpartisexcludedtomaintainrstorder,asthecorrelationpotentialcontainsexpressionsofsecondandhigherorders.ThentheequationforVxcanbediagrammaticallyrepresentedbydiagrams2and3. Forthesecondordereectiveoperatorforthedensitycorrectionwewillusetheexpression( 2{24 ) (2{24) 35
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2{25 ). 2 =4 +4 ++2 +2 ++4 +4 +2 +2 + + +2 +2 ++4 +2 +2 (2{25) Diagrammaticrepresentationofpotential,derivedfromthedensityconditionwillbeusedfortheestablishinginterconnectionwiththefunctionalderivativeapproach. 36
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2.3.1DiagrammaticFunctionalDerivativesForformulationoftheruleswewilluseequation( 2{18 ).Thefunctionneedstobeintroducedaccordingtoequation( 2{26 ): + + + (2{26)Bracketsonthelasttwodiagramsdenotedenominators.ExpressionsfortheenergyinMBPTconsistoflinearcombinationsofterms,whichhaveproductsofmolecularintegralsinthenumeratorsandproductofdierencesofoneparticleenergiesinthedenominators,soitiseasytomakeadiagrammaticrepresentationforsuchexpressions.Totakethefunctionalderivativesfromnumerators,accordingtoequation( 2{18 ),alllinesconnectedtosomevertexmustbedisconnectedfromthecorrespondingplaceoflinking,andwhenthisisdone,anewline(correspondingtooccupiedorunoccupiedorbitals)mustbeinserted.Inthenalstepcontractionwiththecorrespondingfunctionmustbeprovided.Alllinescorrespondingtodenominatorsarestillunchanged.Thisproceduremustbeprovidedforallvertices,becausewhenwetakefunctionalderivativesfromproductsoffunctions,wehaveasumofproducts,accordingtorulesfortakingderivatives.Ifanewdiagramchangesitssign,aminussignmustbeassignedtothisdiagram.Whenfunctionalderivativesfromdenominatorsaretaken,itismoresuitabletouseequation( 2{13 ).Themostgeneralformforthedenominatorcanberepresentedbythefollowingformula: 37
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2{13 )itispossibletowritedownthefunctionalderivativesfromsomedenominatorinthegeneralform: Vs=(K1Xi=1j'i1occj2K1Xj=1j'j1unoccj2)Y;6=1(KXi=1ioccKXj=1junocc)+:::::+:::::+(KXi=1j'ioccj2KXj=1j'junoccj2)Y;6=(KXi=1ioccKXj=1junocc) (2{28) Nowitispossibletowritedownatermwhichincludesthefunctionalderivativefromthedenominator: Vs==XNom(PKi=1j'ioccj2PKj=1j'junoccj2) (PKi=1ioccPKj=1junocc)2Q;6=(PKi=1ioccPKj=1junocc) (2{29) Wenowhavethepossibilitytoformulatehowtotakefunctionalderivativesfromdenominators.Totakefunctionalderivativesfromdenominators,alldiagrams,whereoneofthelinesisdoubledandbetweentheselineswhicharisefromdoubling,thecorrespondingdiagonalfunctionisinserted.Thisproceduremustbeprovidedforallhorizontallinesonthediagramandforallthecontoursthelinescross. 2{17 )isequaltodiagram2,therighthandsiteisequaltodiagram3whenthefunctionalderivativehasbeentakenfromtheexpressionfortheexchangeenergy.Thediagrammaticexpressionforthesecondorderenergyinthegeneralcasehastheform[ 1 ]: +2 +2 (2{30) 38
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2{31 ) + (2{31)Insertingthelinesofunoccupiedorbitalsandmakingcontractionswiththefunction,diagrams4and8ofequation( 2{25 )willbeobtained( 2{32 ): 2 +2 (2{32)Whenfunctionalderivativesaretakenwithrespecttounoccupiedorbitalsandoccupiedorbitalsareinserted,diagrams5and9ofequation( 2{25 )willbeobtained.Whenthisprocedureisdoneforthelowervertex,complexconjugatediagramswillbeobtainedandwewillhavethesamenumberofdiagramsofthissortasintheeectiveoperatorapproach.Whenfunctionalderivativesaretakenwithrespecttooccupiedorbitalsfortheupperandlowervertexesandanoccupiedlineisinserted,wehavethediagrams,givenbyequation( 2{33 ) (2{33)"Brackettype"denominatormeansthedierence"i"k.SummationofthesttwodiagramsaccordingtotheFrantzMillstheorem[ 18 ]andthesameprocedureforthesecond 39
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2{34 ) 2 +2 (2{34)Onthetwodiagramsweimposetherestrictioni6=k.Tomakeadirectcorrespondencewithdiagrams7and11weneedtoadddiagramsthatarisefromtakingfunctionalderivativesfromdenominators( 2{35 ) 2 2 (2{35)Afterthesediagramsareaddedtotheprevioustwo,wewillhaveadirectcorrespondencewithdiagrams7and11.Whenfunctionalderivativesaretakenwithrespecttotheunoccupiedorbitalsandanunoccupiedlineinserted,thesameprocedurewillgiveadirectcorrespondencewithdiagrams6and10ofequation( 2{25 ).Nowconsidertherstdiagramforthesecondorderenergyexpression.Theprocedurefortakingfunctionalderivativesfromtherightsideofthevertex,whichdonotcontainaclosedFringareabsolutelythesameasinthecaseofothersecondandthirddiagramsintheenergyexpression.Whensuchfunctionalderivativesaretaken,wehavefourdiagramscorrespondingtodiagrams1215.TotakefunctionalderivativesfromtheFrings,therstdiagramintheenergyexpressionmustberepresentedwithmoredetail,takingintoaccountthatf=h+JK,infollowingtheform( 2{36 ) = + + +2 +2 +2 (2{36) 40
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2{37 ) 2 +2 +2 +2 (2{37)Thersttwodiagramsareequaltodiagram16;thethirdandfourthequaltodiagrams17and16respectively.Sincethelefthandsiteofeq.12canberepresentedbydiagram3,wehaveanexactequivalencebetweenthefunctionalderivativeandtheeectiveoperatorapproachesintherstandsecondorderforthe"KohnSham"partitioningofthehamiltonian. 2{38 ) 41
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18 ]factorizationtheorem.Sinceallinitialhorizontallinescrossalllines,whichgoupfromlowerlyingvertexes,itispossibletorepresentthediagrams,whichappearafterapplyingtheFrantzMills[ 18 ]theoremintheform( 2{39 ) = + (2{40)Thediagramonthelefthandsideistheresultoftakingfunctionalderivativesfromtypicalenergydiagramsofthirdorder.Applyingthefactorizationtheorem[ 18 ]totherighthandsidewewillhavethesameresultasonthelefthandsideofthisrelation.The 42
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2{41 ): (2{41)Numbersonlinesshowtheconditionalnumberoflineswhichareenteredforsimplicityoffurthermanipulations.Thedesirablesumofdiagramshastheform( 2{42 ) + +:::::::::+ (2{42)Togetherwithdiagramswhichcorrespondtofunctionalderivativesfromdenominators,thissetofdiagramsformsproductslike(k)+(l),wherek+lequalstheorderofMBPT. 43
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2{43 ) (2{43) IntheseexpressionsNommeansnumerator,Aimeanstherestoftheterms,whicharepresentindenominators.Aftermultiplicationanddivisionby(kl)wehave( 2{44 ) (kl)(l+A1)(k+A1)::::(k+Am)+XNom(kl+A2A2) (kl)(l+A1)(l+A2)(k+A2)::::(k+Am)+::::::::+XNom(kl+AmAm) (kl)(l+A1)::::(l+Am)(k+Am)==XNom (2{44) Aftercancellationofallequivalentterms,wehavetheexpressions( 2{45 ) 44
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2{46 ) + (2{46)Namelythesetwodiagramswillappearaftertakingfunctionalderivativesusingdiagrammaticrulesfortakingfunctionalderivatives.Proofforthecasewherefunctionalderivativesaretakenwithrespecttounoccupiedorbitalsisthecompleteanalogofthisone.Hence,wecanformulateasecondstatement.Statement2:Whenfunctionalderivativesfromsomevertexofadiagramaretakenwithrespecttooccupied(unoccupied)orbitals,andalinewhichcorrespondstooccupied(unoccupied)orbitalinserted,togetherwithdiagramswhicharisefromtakingfunctionalderivativesfromdenominators;wealwayshavediagrammaticexpressions,whichcorrespondstopartofPnk=0((k)+(nk))Sc.Thisstatementisadirectcorollaryoftheaboveprovedstatementabout"brackettype"denominatorsandstatement1.Sincewehavecorrespondenceinthesecondandnthorder,usingthemethodofmathematicalinduction,itispossibletoprove,thatthecorrespondencetaksplaceinallorders(niteorinnite). 45
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19 ]toexctractthecorrespondingexchangecorrelationpotential.SincethefullCIenergydoesnotdependuponthechoiceoforbitalbasisset,theOEPprocedure[ 13 15 ]cannotbeuseddirectlyforthiscase.Theinnitesumofallenergycorrectionsdoesnotdependuponchoiceoforbitalbasisset,buteachtermofthissumdoesdependuponthechoiceoforbitals.Thisfactenablesustoconsideralltermoftheinnitesumofenergycorrectionsasorbitaldependentfunctionals.AftertheconstructionofthesetV(1)xc::::V(1)xcandsummationuptoinnity,wewillhavetheexchangecorrelationpotentialwhichcorrespondstothefullCI.Usingtheequivalenceofthefunctionalderivativeandtheeectiveoperatorapproach,itispossibletoconcludethataftersummationofthissetofpotentials,weagainwillhavethesameresultasintheeectiveoperatorapproach.Redenedinsuchaway,theOEPprocedure[ 13 15 ]forthefullCIenergyproducesthesamedensityastheZMPmethod[ 19 ]wouldfromfullCI. 46
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20 ]andimplementedformoleculesbyHirataet.al[ 21 ].NumericalresultsinthatworkshowareasonabledescriptionofbothvalenceandRydbergexcitedstates,partlyduetothecorrectasymptoticbehavioroftheexchangepotential,butalsoduetotheeliminationoftheselfinteractionerror.Inparticular,OEPTDDFTissuperiortostandardapproacheslikeTDDFTbaseduponlocaldensityapproximation(LDA)orBeckeLeeYangParr(BLYP)functionals.Similarly,exchangeonlyOEPwithexactlocalexchange(EXX)[ 22 23 ]hasbeenshowntogreatlyimprovebandgapsinpolymers[ 23 ]AnotheradvantageofOEPbasedmethodsisthatsincevirtualorbitalsintheexchangeonlyDFTaswellasoccupiedorbitalsaregeneratedbyalocalpotential,whichcorrespondstotheNparticlesystem,thedierencesbetweenorbitalenergiesofvirtualandoccupiedorbitalsoeragoodzerothorderapproximationtotheexcitationenergies[ 16 17 ].ThisisnotpossibleinthecaseofHartreeFocktheory,whereoccupiedorbitalsaregeneratedbyanN1particlepotentialandtheenergiesofunoccupiedorbitalscomefromNelectronpotential,andthereby,approximateelectronanities.OnceOEPcorrelationisadded[ 13 17 ],theessentialnewelementinthetimedependentDFTschemeistheexchangecorrelationkernel,whichintheadiabaticapproximationisdenedasafunctionalderivativeoftheexchangecorrelationpotentialwithrespecttothedensity[ 20 ]: 16 17 ]thishasbeencalledOEPKohnSham(KS)todistinguishitfromotherchoicesforH0.Itis 47
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14 24 ].OnlyforthischoiceisthereanimmediatecorrespondencebetweenthefunctionalderivativeandKSdensityconditionthatasingledeterminantprovidestheexactdensity[ 24 ].TheotherchoicesforH0leadtowhatiscalledabinitioDFT[ 16 17 ]andbecomesabetterseparationofhamiltonianintoH0+V0.ItsconvergencefororbitaldependentperturbationapproximationstothecorrelationenergyfunctionalsareconsiderablybetterbehavedcomparedtotheKSchoicethatfrequentlycausesdivergence.Nevertheless,inthisrstapplicationwewilladheretothestandardKSseparation,wherethepotentialandkernelarefunctionalderivatives.Thetraditionalwayofderivingkernelsandpotentialsistotediouslyderivealltermswiththeuseofthechainruleforfunctionaldierentiation.Yet,eventheexchangeonlykernelhasacomplicatedstructureinOEP,anditsfurtherextensiontoincludecorrelationwouldbealmostimpossible.Toavoidtheuseoftraditionalmethods,aneectivediagrammaticformalismfortakingfunctionalderivativeshasbeendevelopedandimplemented. 3.1.1FormalismTheadiabatic(frequencyindependent)approximationtothekernelofthenthorderisdenedas( 3{2 ) 13 ]( 3{3 ) 48
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3{3 )into( 3{2 )weobtain 'p(r4)(Zdr5E(n) Thismayberewrittenas( 3{5 ) Vs(r3)(Zdr5E(n) Equation( 3{5 )canbemademoreexplicit, Vs(r3)[Zdr5E(n) Usingthefactthat[ 14 ] 3{6 )inthefollowingform( 3{8 ) 3{3 )wehavetheexplicitexpression( 3{9 )forh(r3;r6) Vs(r3)[Xp;q6=pZdr5E(n) 49
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3{9 )willbeusedasthebasisforthediagrammaticprocedurefortakingfunctionalderivativesfromtheOEPMBPTexchangecorrelationpotentials.Thersttermofequation( 3{9 )containsthefunctionalderivativewithrespecttothepotential,fromanexpressionwhichalreadyincludesthefunctionalderivativeoftheenergywithrespecttothesamepotential.Diagrammaticrulesfortakingfunctionalderivativesfromnumeratorsofdiagramsarethefollowing: 2{26 )togetthefullyconnecteddiagram 2{26 )shouldbeinsertedbetweendoubledlinesforalllineswhichareintersectedbythedenominatorline.Thisprocedureshouldbedoneforallpossibledenominatorlines.TherulesofinterpretationarethesameasfortheusualGoldstonediagrams,exceptthenumericalfactorshouldbetakenfromtheinitialdiagramwhenfunctionalderivativesaretaken.Aftertakingthefunctionalderivativesfromtheexchangecorrelationwithrespecttothepotentialusingdiagrammaticrules,asetofdiagrams,containingthe^functionvertexwillappear.Togettherstelementofequation( 3{9 ),itisnecessarytoapplydiagrammaticrulestothesetoffunctionalderivativediagramsonemoretime.Afterthatwewillhaveasetofdiagramscontainingtwo^functions,whicharenecessaryfortheconstructionoftheexchangecorrelationkernels.Takingintoaccountequation( 2{14 ),thesecondtermofequation( 3{9 )canberewritteninthefollowingway( 3{10 ) Vs(r3)['i(r5'a(r5)'i(r6)'a(r6) 50
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3{10 )canberepresentedbydiagram(V)( 3{11 ) (3{11)Thefunctionalderivativesfromdiagram(V)producethesetofdiagrams(V1)(V6)ofset( 3{12 ) (3{12)Duringthederivationofdiagrams(V1)(V4),diagramscontaining"brackettype"denominatorswillappear,becauseoneofthestepsinthediagrammaticrulesofdierentiationrequiresdetachingtheunoccupiedlinefromdiagramVandinsertingthelasttwopartsofthe^function( 2{26 ).Suchdiagramscanbetransformedintoasetofregulardiagrams,usingthediagrammaticrelation( 3{13 ).Thisrelationshowsusthatforthetransformationtoregulardiagramsitisnecessarytodoubleeachofthedenominatorlinesandinsertthelasttwomembersofthe^function( 2{26 ),subjecttotherestrictedsummation,l6=k.Weusethesameprocedureforthecaseofoccupiedorbitals. + = + +...+ (D1)(3{13) 51
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3{9 )anddiagrams(V1)(V6)itispossibletobuildtheexchangecorrelationkernelstoanyorder. 20 ],andthenrederivedandinitialyimplementedbyHirataet.al[ 21 ].Suchaderivationrequiresmucheort.Hereweoerafacilederivationwiththediagrammaticformalism.Theexchangeenergy(Ex=1 2Pi;j)canberepresentedbydiagram( 3{14 ). (EX)(3{14)AftertakingtherstfunctionalderivativewithrespecttoVswewillhavethediagram( 3{15 ). (3{15)Afterthatthediagrammaticrulesmustbeappliedonemoretimetogetthesecondfunctionalderivative,asisnecessaryaccordingtoequation( 3{9 ).Aftertakingthesecondfunctionalderivativeswewillhavediagrams(FX1)(FX8)oftheset( 3{16 ) (3{16)Aftertheadditionofdiagrams(V1)(V6)wewillhaveallthediagramsnecessaryforbuildingtheexchangeonlykernel.Duringtheinterpretationofdiagrams(V1)(V6)the 52
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3{10 ).Interpretationofdiagrams(FX1)(FX8)and(V1)(V6)givesustheexpression( 3{17 ) ("i"a)("j"b)2Xi;j;a;b'i(r3)'a(r3)'j(r6)'b(r6) ("i"a)("j"b)+2Xi;j;k;a['j(r3)'i(r3)'i(r6)'a(r6)+'j(r6)'i(r6)'i(r3)'a(r3)] ("i"a)("j"a)2Xi;j;a;b['a(r3)'b(r3)'b(r6)'i(r6)+'a(r6)'b(r6)'b(r3)'i(r3)] ("i"a)("i"b)+2Xi;j;k;a'i(r3)'a(r3)'j(r6)'a(r6) ("i"a)("j"a)2Xi;j;a;b'i(r3)'a(r3)'i(r6)'b(r6) ("i"a)("i"b)+2Xi;j;a'i(r3)'a(r3)'j(r6)'a(r6) ("i"a)("j"a)2Xa;b;i'i(r3)'a(r3)'i(r6)'b(r6) ("i"a)("i"b)+2Xi;j;a['i(r3)'j(r3)'j(r6)'a(r6)+'i(r6)'j(r6)'j(r3)'a(r3)] ("i"a)("j"a)2Xa;b;i['a(r3)'b(r3)'b(r6)'i(r6)+'a(r6)'b(r6)'b(r3)'i(r3)] ("i"a)("i"b) (3{17) Expression( 3{17 )forh(r3;r6)exactlycorrespondstotheexpression,obtainedbyHirataet.al,butthediagrammaticderivationrequiresfarlesseortandisunambiguousintermsofsignsandnumericalfactors.Sinceallthediagramscontainonlyonecontour,itispossibletomakesummationsinequation( 3{17 )onlyoverspatialorbitals,and,asaresultf(r1;r2)andf(r1;r2)willappearseparately.Theexchangeonlykerneldoesnotcontainthef(r1;r2)part,whichisacriticaldierencebetweentheexchangeandcorrelationkernels. 53
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3{18 ). (3{18) Diagrams(VC1),(VC2),(VC5),(VC6)and(VC15)of( 3{18 )haveanexternalfactor4,diagrams(VC9)and(VC10)havenofactor,whiletherestofthediagramshaveafactorof2.Togetthecorrelationkerneldiagramsweneedtotakefunctionalderivativeswithrespecttothepotentialfromdiagrams(V1)(V15).Afterapplyingthediagrammaticrulestodiagram(VC1)andtheuseofrelation( 3{13 ),diagrams(VC11)(VC116)ofsets( 3{19 )and( 3{20 )willappear. (3{19) 54
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Dierentiationofdiagram(VC2)producediagrams(VC21)(VC216)ofset( 3{21 ).Thesediagramshavethesameskeletonstructure,butdierentpositionsofindices. (3{21) Aftertakingthefunctionalderivativesfromdiagram(VC3)diagrams(VC31)(VC314)( 3{22 )willappear. (3{22) 55
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3{23 )willappear. (3{23) Diagram(VC5)producesdiagrams(VC51)(VC516)( 3{24 )afterdierentiation. (3{24) Dierentiationofdiagram(VC6)generatediagrams(VC61)(VC616)( 3{25 ), (3{25) 56
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3{26 ). (3{26) Finally,diagram(VC8)producesdiagrams(VC81)(VC814)( 3{27 ). (3{27) Inthesamewaytherestofthediagramsshouldbedierentiated.SincetheFockoperatordependsuponoccupiedorbitals,Fringsonthediagrams(VC9)(VC15)mustalsobedierentiated.Diagram(VC9),aftertakingfunctionalderivatives,producesdiagrams(VC91)(VC910)( 3{28 ). (3{28) 57
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3{29 ). (3{29) Diagram(VC11),producesdiagrams(VC111)(VC1116)( 3{30 ). (3{30) 58
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3{31 ) (3{31) Dierentiationofdiagram(VC13)givesusdiagrams(VC131)(VC1315)( 3{32 ) (3{32) 59
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3{33 ) (3{33) Thelastsetofdiagrams(VC151)(VC1515)canbeproducedbydierentiationofdiagram(VC15)( 3{34 ) (3{34) Togetherwithdiagrams(V1)(V6),alldiagramspresentedinthissectionformaset,whichisnesessarytoconstructthecorrelationkernel.TheinterpretationofalldiagramsisgiveninAppendix A 60
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(3{35) Diagrams,whichcontainonlyonecontourcannotproducehcomponentsafterdierentiation.Threeofthediagrams(VC5)(VC8)afterdierentiationcanproduceonlydiagramscontributingtothehorhparts.Thediagrams(VC91)(VC97),(VC99),(VC910),(VC101)(VC107),(VC109),(VC1010),(VC111)(VC1110),(VC1113)(VC1116),(VC121)(VC1210),(VC1213)(VC1216),(VC131)(VC1312),(VC1314),(VC1315),(VC141)(VC1412),(VC1414),(VC1415)havethesameproperty.Diagramscontaining2ormorecontourscanproduceallspincomponents.Setsofdiagrams(VC17)(VC116),(VC27)(VC216),(VC37)(VC314),(VC47)(VC414),(VC157)(VC1512)havetwocontours,butboth^functionspresentinoneofthecontoursmeansthatthesediagramsmakeacontributiontohorhparts,buthaveanextrafactorof2.Thatfactorappearsaftersummationoverallspinorbitalsofthesecondcontour.Diagrams(VC11)(VC16),(VC21)(VC26),(VC31)(VC36),(VC41)(VC46),(VC98),(VC108),(VC1111),(VC1112),(VC1211),(VC1212),(VC1313),(VC1413),(VC151)(VC156),(VC154)(VC1515)have^functionsondierentcontours,sotheycontributetobothhandhparts.Diagram(VC1513)haveanadditionalfactorof2andalsocontributeintobothspinparts.Tobuildallspinpartsofthecorrelationkernelweneedtosubstitutetheabovediagramsintoequation( 3{35 )Thenextessentialpropertyoftheexchangeandcorrelationkernelsisthesymmetrywithrespecttopermutationofitsarguments 61
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3{35 )and 3{35 )weagainneedtoanalyzethestructureofthediagrams.Diagrams(VC12),(VC15),(VC17),(VC21),(VC26),(VC27),(VC52),(VC57),(VC59),(VC62),(VC67),(VC69),(VC116),(VC126),(VC1314),(VC1415),(VC1513),(VC1511),(VC1512),(V5)and(V6)aresymmetricwithrespecttothepermutationofvariablesr1andr2.Afterconsiderationoftherestofthediagrams,itiseasytoseethatforeachdiagramthereisanotherdiagram,whichdiersfromtheinitialonebyonlytheinterchangedvariables,r1andr2.Suchpairsprovideinvariancewithrespecttopermutationofvariables.DuringtheconstructionofthecorrelationkernelKohnSham(KS)orbitalsandorbitalenergiesareused.ThisimpliesthefactthatourzeroorderhamiltonianischosentobetheKohnShamchoice 1{65 )shouldbesolved.IntheadiabaticapproximationithastheRPAlikeform.Thecriticalnewquantityistheexchangecorrelationkernel,f(xc);whichcanbeseenfromtheforegoingisaquitecomplicatedquantityifweinsistuponobtainingitrigorouslyforOEPMBPT(2).ThediagonaldependenceofAonaitellsusthatifwearetogetgoodexcitationenergies, 62
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16 ]whenaccurateexchangecorrelationpotentialsareused.Inthissense,theorbitalenergiesinKSDFTshouldhaveacertainmeaning.Furthermore,aspointedoutinref[ 16 ],wecanalsoconsiderthisequationtooeraKoopmanslikeapproximationtotheprincipalionizationpotentials,since,barringpathologicalbehavior,whenweallowanelectrontobeexcitedintothecontinuum,itsorbitalawillthenhavenooverlapwiththeboundorbitalmatrixelementsofhKSandthekernel.Consequently,weareleftwithnothingbutiintheTDDFTequations.Sointhis'sudden',adiabaticapproximation,theKSorbitalenergiesshouldoeranestimateforeachoftheprincipalIp's,notjustthehighestoccupied(homo)one.WhenbasedupontherelativelycorrectVxcobtainedfromabinitiodft[ 16 17 ]thisestimateissuperiortoKoopmans'theoremforthehomoandtherstfewvalenceIp's,butisinferiorforthecoreorbitals[ 26 ].SeealsoChong,etal[ 25 ].However,theOEP2semicanonical(sc)abinitiodft[ 16 17 ]approximationhasthedistinctadvantagethatitusesamuchbetterbehavedunperturbedHamiltonianthantheusualKSchoice,H0=PihKS(i).ToillustratetheevaluationofthekernelandthesolutionofTDDFTequations,weconsidertheNeatom.AlltheexcitedstatesinNecorrespondtoRydbergexcitedstates.Toobtainreasonablevaluesrequiresaquiteextensive,diusebasisset.WechoosetostartwiththeROOSATZPatomicnaturalorbitalbasis[ 27 ]consistingof(14s9p4d1f)primitivegaussianfunctionscontractedtoa[5s4p3d1f]set.Thisunderlyingbasiswasthenaugmentedbyasetofeventempereddiusefunctions[3s3p3d],withexponentialparameters=abn;as=0:015;ap=0:013;ad=0:012;b=1/3.Theauxiliarybasisischosentobethesame,butwithoutthepandddiusefunctions,sinceforthedescriptionoftheexchangecorrelationpotential,diusefunctionsarenotnecessary.TheorbitalenergyestimatesareshowninTable 31 .Theionizationpotentialequationofmotioncoupledcluster(IPEOMCC)resultinthisbasisfortheNehomoIpis21.3eVwiththeexperimentalvaluebeing21.5645 63
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Orbitalenergiesandzeroorderapproximationstoexcitationenergies ExchangeonlyOEPMBPT(2)orbitalIP[ 28 ]Orbital"a"HOMOOrbital"a"HOMOExcitationenergiesenergiesenergies(exp) 2s48.4246.1943.252p21.5622.5120.233s5.1517.367.5812.6516.853p2.5919.923.7216.5118.704s1.9520.562.6617.5718.733d1.4921.022.0518.1818.97 [ 28 ].ThiscomparestotheOEPxvalueof22.51andOEPMBPT(2)valueof20.23.Theremainingunoccupied,butnegativeenergy3sorbitalischangedbyover2eVduetotheMBPT(2)correlation.TheOEP2(sc)abinitodftvaluechangesthisto5.18eV,attestingtothepoorconvergenceofthestandardKSpartitioningoftheHamiltonian.OnceOEP2(sc)calculationsaredoneforthe3p,4s,and3dstates,thereissimilaragreementbetweentheOEPxandOEP2(sc)results,contrarytothoseshowninTable 32 forthestandard(ks)choice. Table32. ExcitationenergiesofNeatomusingOEPMBPT(2)KohnShamorbitalenergies TermEOMCCSDTDDFTexchangeonlyTDDFTOEPMBPT(2)Exp[ 29 ]. Despitethedierences,atleastboththeexchangeonlyandtheOEPMBPT(2)givequalitativelycorrectresultsfortheRydbergseries.Helpingtoensurethisisthefactthattheexchangepotentialhasthecorrectasymptoticbehaviorandtheexactcancellation 64
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30 ].WhenwesolvetheTDDFTequationswiththekerneldevelopedinthispaper,weobtaintheresultsforexcitationenergiesshowninTable 32 .Ingeneral,justasweobservedfromtheorbitalenergydierences,theOEPMBPT(2)resultstendtofallonthelowsideofexperimentandtheOEPxonthehighside.Infact,exceptfortheexcitationenergyforthelowest1Pstate,wheretheveryloworbitalenergyof7.58biasestheresults,anaverageofthetwowouldseemtobeaboutright.Thereare,however,stilldiciencesinthebasisset,asseenbytheEOMCCSDresultscomparedtoexperiment,wherebesidestheadditionalcorrelationeectsintroducedbyEOMCCSDT[ 31 ],thefurtherextensionofthediusefunctionswouldremovethe~1eVerrorinthehighestlying(1S)state.Ofcourse,thedependenceofTDDFTonthebasisandthatforatwoparticletheorylikeEOMCCshouldbequitedierent.ThegreatsensitivityoftheresultstotheorbitalenergiesfromtheunderlyingKSDFTcalculationcanbefurtherappreciatedbysimplytakingtheenergiesfromtheOEPxresultsandusingthemintheevaluationofthekernelandthematrixelementsintheTDDFTequations.TheseresultsareshowninTable 33 .ThenearcoincidenceofresultsforOEPxandOEPMBPT(2)isapparent,withbothnowbeingtoohigh.AsimilarexperimentcanbemadewhereweuseorbitalsandorbitalenergiesfromOEP2(sc)abinitodftresultsasdescribedelsewhere[ 16 ],toobtaintheresultsinTable 34 .ThatistheTDDFTequationsandthekernelareassumedtobethesame,butweuseorbitalsandorbitalenergiesobtainedfromOEP2(sc).HerewealsoshowtheresultsfromstandardTDDFTapplicationsusingtheLDAandB3LYPfunctionalsforcomparisonpurposes.Clearly,wehaveimprovedresultsatboththeOEPx(sc)andOEP2(sc)levels,withthesamepatternoftheformerbeingtoohigh,butlessso;andthelatter,toolow,butbetterthanbefore.Itisapparentthattheproperwaytoachievethebenetsof 65
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Table33. ExcitationenergiesofNeatomusingexchangeonlyorbitalenergies TermTDDFTexchangeonlyTDDFTOEPMBPT(2)Exp[ 29 ]. Table34. ExcitationenergiesofNeatomusingorbitalenergiesandorbitalsfromOEP2(sc).AllequationsforTDDFTarethesame TermEOMCCSDB3LYPLDATDDFTTDDFTExp[ 29 ].[ 32 ][ 32 ]exchangeOEPonly(sc)MBPT(2)(sc) ThecornerstoneofOEP(sc)forexchangecorrelationpotentialsisthatwecanimposetheconditionthattheKSdeterminanthastogivetheexact,correlateddensity.ThenthisdensityconditionprovidesequationsthatdeneVxcforagivenfunctional[ 17 ].Sincenoenergyvariationalconditionorfunctionalderivativeisused,wecanchangethechoiceofH0fromtheKSonetothesemicanonical(sc)onethatoersgreatlyenhancedconvergenceofperturbationtheory.ButthenthereisnoapparentimmediatecorrespondencetoafunctionalderivativeasthereisinthestandardKStheory.Asimilar 66
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16 17 26 ],theradiusofconvergenceispoor,causingpoorconvergence[ 16 ]indeterminingthecorrelationpotential.Nevertheless,therststepintheunderlyingframeworkhasnowbeendenedtoapplyTDDFTwithOEPcorrelationpotentials.However,thecomplexityisgreat,thoughimpositionofadditionalrigorousconditionsmightresultinsimplicationofthekernel.Lackingsuchsimplications,thequestionarisesaswhetherthisisacaseofdiminishingreturnstoevaluatesome203diagramstoretaintheattractive,oneparticle,correlatedstructureofDFT?First,theanswersforNearenotverygoodattheOEPMBPT(2)level.Furthermore,thebasissetdependenceofOEPmethodswhendoneingaussianbasisisseveretoevengettheVxcright[ 16 ].ThefailureofmostsuchOEPcalculationstosatisfytheexactHOMOcondition=isacaseinpoint[ 16 26 ]. 67
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31 ])arelesstimeconsumingthanisthepresentcalculation.Aslongasthekernelforarigorousorbitaldependentcorrelationpotentialisthiscomplicated,twoparticlewavefunctiontheorieslikeEOMCCarebothsuperiorandeasiertodo.ButbeforewecanfurtherexploittheinterfacebetweenDFTandwavefunctiontheorytothebenetofboth,itisrequisitetohavetheorbitaldependentkerneldened.This,anditsinitialevaluation,iswhatthischapteroers. 68
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^(r)=Xp;q<'pj(rr1)j'q>a+paq(4{1)UsingWicktheorem,werewritethisoperatorinnormalformwiththeKohnShamdeterminantastheFermivacuum: Thesecondtermisks,hence,thersttermwillbecalledthedensitycorrection: (r)=Xp;q<'pj(rr1)j'q>fa+paqg(4{3)SincetheconvergedKohnShamschemegivesanexactdensity,allcorrectionstothisdensitymustbeequaltozero.Hence,ifweconstructaneectiveoperatorofthedensityusingMBPT,thecorrectiontothedensitymustvanishinanyorder[ 17 24 ].TherstandsecondordersoftheMBPTdensityconditioncanbewritteninthefollowingway (4{5) UsingtheKSpartitioningofthehamiltonian[ 13 ]andtakingintoaccounttherelationfpq="ppq+<'pjVHFxVx()Vcj'q>,itispossibletoderivethesameexpressionsfor 69
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13 33 ].However,thisisanumericallyhopelessprocedurewithoutanyresummationofterms:(1)itsuersfromaddingalargediagonaltermintotheperturbation;(2)MBPT(2)isnotboundedfrombelowandanyeorttousethevariationalconditionE(2)c However,toavoiddealingwithnondiagonalH0,asemicanonicaltransformationwillbeperformedtoobtainthemoreconvenientzerothorderhamiltonian, (fii+fjjfaafbb)(fifc)2;XXi;j;k;a;b()'k(r1)'a(r1) (fii+fjjfaafbb)(fkkfaa);XXi;j;k;a;b()'k(r1)'i(r1) (fii+fjjfaafbb)(fkk+fjjfaafbb) 70
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(fii+fjjfaafbb)(fii+fjjfccfbb)+Xa;b;ifibfia'i(r1)'a(r1) (fiifaa)(fiifbb)Xi;j;afajfij'i(r1)'a(r1) (fiifaa)(fjjfaa)+Xa;b;ifiafib'a(r1)'b(r1) (fiifaa)(fiifbb)Xi;j;afaifaj'i(r1)'j(r1) (fiifaa)(fjjfaa)Xi;j;a;bfbj(+)'i(r1)'a(r1) (fiifaa)(fjjfbb))X1(r1;r) (4{9) where 13 17 ]. 4.2.1TotalEnergiesGroundstateenergiesarecalculatedforseveralsystemswithresultspresentedinTable 41 .TheuncontractedROOSATZPbasisisusedfortheNeatom,whiletherestoftheatomsandmoleculesarecalculatedusingtheuncontractedROOSADZPbasis[ 27 ].Allthemoleculesandtheirionsareconsideredtobeintheequilibriumgeometry[ 36 ]ofthecorrespondingneutralsystem,exceptwhenpotentialenergycurvesareconsidered.ForabinitioDFTcalculationsexchangecorrelationpotentialsareusedfrom[ 13 ]andequations( 4{8 )( 4{9 ).ForthecomparisonenergiesobtainedfromKSDFTwiththePerdewBurkeErnzerhof[ 34 ](PBE)exchangecorrelationpotentialandcoupledclusterwithsingle,doublesandperturbativetripleexcitation[ 35 ](CCSD(T))arealsoshowninTable 41 .Totalenergies,calculatedwithOEPMBPT(2),basedonthedensityconditionapproachwiththesemicanonicalpartitioningaremuchclosertoCCSD(T)thenOEPMBPT(2)withtheKSpartitioning.Thelattergreatlyoverestimatescorrelationenergy 71
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37 ]tothevariationalOEPMBPT(2)fortheKSpartitioningithasthesameproblems.However,inthecaseofthenonvariational,semicanonicaldensityconditionapproach,theunboundnessfrombelowislessofaproblem.ThesemicanonicalchoiceofH0providesgoodapproximationstotheenergyandwavefunctioninMBPT(2)andthecorrespondingtotalenergiesaremuchclosertohighlyaccurateCCSD(T)ones,atleastinthechosenbasis.ThecomputationalcostoftheOEPMBPT(2)methodiscomparablewiththecostofMBPT(2),thescalingofbothmethodsisN5,whereNisthenumberofbasisfunctions.ThisismoreexpensivethanconventionalDFT(scaleslikeN3),butlessexpensivethenCCSD(T),computationaltimeofwhichisproportionaltoN7.Thecorrelationpotential, Table41. Totalenergies MP2OEPKSOEPsemiPBECCSD(T) calculatedwiththesemicanonicalOEPfortheNeatom[ 16 ]isclosetothecorrespondingquantumMonteCarlopotential,andwecanexpectthatforothersystems,whereQMCisnotavailable,semicanonicalOEPMBPT(2)potentialsshouldbeagoodalternative.FortheopenshellcasepotentialsproducedbyKohnShamOEPshowanoverestimationofthecorrelationenergy,thesamesituationpreviouslyreportedfortheclosedshellcase[ 16 ].(Fig 41 and 42 ).Thecorrect1 72
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41 and 42 )isguaranteedbytheuseoftheColleNesbetseedpotentialalgorithm[ 30 ].InallourcalculationsaSlaterpotential[ 38 ]wasusedasaseedpotential,sincetheColleNesbetalgorithmrequirestheseedpotentialtobeascloseaspossibletotheactualpotential.TwoseparateSlaterpotentials,oneforalphaandtheotherforbeta,wereused.ForalltheclosedshellsystemsfromTable 41 ,thenumerical Figure41. ExchangeandcorrelationpotentialsofLiatom(radialpart).A)Exchangepotential.B)Correlationpotential Figure42. ExchangeandcorrelationpotentialsofO2moleculeacrossthemolecularaxis.A)Exchangepotential.B)Correlationpotential resultsobtainedfromOEPMBPT(2)withthesemicanonicalpotentialandfromDFT 73
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42 .InexactDFT,theenergyofthehighestoccupiedmolecular Table42. Ionizationpotentials(ine.v.) HOMO,OEPsemiE,OEPsemiHOMO,PBEE,PBEExp Ne21.0121.5913.3521.6921.56N216.8915.0310.2715.4115.58CO13.6813.699.0513.8714.01CN4.193.510.1493.723.86H2O12.3712.737.0812.4612.62N15.2514.528.3014.7314.53Li4.895.383.235.595.39 orbital(HOMO)correspondstothenegativeoftheexactverticalionizationpotential.InthecaseofthesemicanonicalOEPtheHOMOenergyisclosetothecorrespondingEvalues,whatcanbeexplainedbythecorrectshapeoftheexchangecorrelationpotential(Fig 41 and 42 andref[ 16 17 ]),whilePBEdoesnotexhibitcorrectbehaviorofpotentialandthus,failstoreproducethecorrectHOMOenergyvalues.FailuretoreproducethecorrectHOMOenergycausestheincorrectintegerdiscontinuityofthePBE(oranyotherGGA)exchangecorrelationpotential[ 39 40 ]. 74
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43 .PotentialcurvesfortheLiH,OHandHFmoleculesareshownonFig 43 44 and 45 .ForallcurvesthedissociationenergieshaveapproximatelythesamelevelofaccuracyasintheMBPT(2)case,butthesemicanonicalOEPMBPT(2)improvestheshapeofthecurves. Table43. Dissociationenergies(inkJ/mol) PBEOEPsemiExp. 41 42 ]arereported,usingtheuncontractedROOSADZPbasissets:13s9p3dforcarbonand8s4p1dforhydrogen.Equilibriumgeometriesforbothstatesaretakenfromreference[ 41 ].Theenergiesforthetwostates,calculatedwithdierentmethodsarepresentedinTable 44 .Therearesignicantdierencesintheenergyseparationcomparedtotheexperimentalvalue(8.998kcal/mol)forallofthepresentedmethods.Thisisabasissetissue,butcanalsobeexplainedbythefactthatthesingletstateofmethylenehasasignicantcontributionfromtwodeterminants[ 41 ],soMBPT(2)isamuchpoorerunderlyingapproximationfortheirdierencethaninniteorderCCSDorCCSD(T),andespeciallythetwodeterminantCCSD(TDCCSD)results[ 41 ].WheretheabsolutevaluesofPBEenergiesarenottoogoodforthetwostatesofCH2,thedierenceisconsistentwithMBPT(2)anditsOEPgeneratedabinitiodftform. 75
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Singletandtripletenergiesofmethylene CCSD(T)39.12700439.10975110.826CCSD39.12341239.10456911.824OEPsemi39.10399439.07915215.588PBE39.10832139.08344815.610MP239.10256539.07767415.619 76
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LiHpotentialenergycurve. 77
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OHpotentialenergycurve. 78
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HFpotentialenergycurve. 79
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5 6 ](DFT)isawidelyusedmethodforgroundstateenergiesandotherproperties.ThetimedependentextensionofKohnShamDFT,whoserigorousfoundationhasbeenquestioned[ 43 ],isaverypopularmethodforthedescriptionofexcitedstatesenergies,sinceitcanoftenprovideaccurateresultsbythediagonalizationofasingleexcitationdimensionalmatrixasintimedependentHartreeFockormonoexcitedCI.HoweverKSDFTandTDDFThavetheirnaturallimitations.StandardfunctionalssuerfromanincompletecancellationofselfinteractiontermsintheKSequationsandincorrectlongrangeasymptoticbehaviorofthepotentials.Bothcanbeparticularlytroublesomeforexcited,particularlyRydbergstates,andionizedstates[ 16 21 48 ],makingitdiculttogetsystematicallyimprovableresultsfromstandardTDDFT.ExactexchangedensityfunctionaltheoryisbasedontheoptimizedeectivepotentialmethodintroducedbyTalmanandShadwick[ 15 ].InthismethodtheexchangepotentialisdenedasthefunctionalderivativeofthenonlocalorbitaldependentexchangefunctionalfromtheHartreeFockmethod(EXX),ieEX=(1)=VX(1):TheOEPmethodisfreefromtheselfinteractionproblem,itexhibitsthecorrect1 44 ] 16 22 45 { 47 ],andthesealternativeapproachescansuerfromnumericalproblems.However,whenevaluatedwithapropertreatmentofthedensitydensityresponsematrixX,mostofthenumericalproblemsthathavebeenencounteredareresolved. 80
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14 ],butessentiallymodiedbyBartlettetal[ 16 17 ]toavoidthefailuresofasimplesumofKSoneparticleHamiltoniansastheunperturbedproblem,togeneratecorrelationpotentials.Thisisthecornerstoneofabinitiodft.Theconceptualdierenceisthatthedensityconditiondoesnotexplicitlyusethevariationaldetermination,EXC=(1)=VXC(1).Thisdierenceisofcriticalimportanceingeneratingcorrelationpotentialsfromlowordersofperturbationtheory.Withoutthatchange,noloworderorbitaldependentcorrelationfunctionallikethatfromMBPT2willgenerallywork,butwiththosechanges,itdoesverywell[ 16 48 ].Thosemodicationstotheperturbationtheoryalsopertaintotheexchangeonlycase,thesubjectofthispaper,butforthatproblemthedistinctionsarelessimportant[ 16 ].Oncefollowingthisapproach,whichstartswiththeKSchoiceofH0;thedistinctionsbetweenapplyingthedensityconditionandusingdirectfunctionaldierentiationismoreconceptualthanessential,asthereisacorrespondenceinanyorderofperturbationtheory[ 24 37 ].Thedirectoptimizationprocedureadvocatedbysome,builtupontheabovevariationaldeterminationofthefunctionalderivative,thoughformallyequivalent,diersfromtheoriginal,XbasedOEPinthedetailsofimplementation.However,aswaspointedoutbyStaroverovet:al[ 45 ]undercertaincombinationsofmolecularandauxiliarybasissets,particularlywhenthelatter'sdimensionislargerthanthatfortheformer,thedirectoptimizationmethodcangivetheHartreeFockenergyanddensity.Thelattercanbeviewedasatrivialsolution,asitcanbeshowntocorrespondtothesolutionofaweightedleastsquaresexpression.Ontheotherhand,theXbasedOEPwithproperhandlingis 81
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49 ]observestheoccurrenceofadegeneracyinperformingOEPcalculations,andtoinvestigatethis,intentionallyconstructaproblemwheretheHOMOandLUMOorbitalsoftheKSproblemaredegenerate.Subjecttodirectminimization,theyproceedtoreportonthefailureoftheOEPprocedure.Ontheotherhand,itisshowninthispaperthattheoriginalXbasedapproachhandlesthisproblem,too.TheexchangeonlytimedependentOEPhasbeenconsideredbyGoerling[ 20 ]andimplementedbyHirataet.al[ 21 50 ].However,allresultstodateareforclosedshellsystems.Tofurtheraddresstheexactlocal(OEPx)exchangeversustimedependentHartreeFock(TDHF),TDOEPxisgeneralizedtotreatexcitedstatesforopenshellspecies.ResultsfromadiabaticTDOEPxareingoodagreementwithTDHFforbothexcitationenergiesandpolarizabilities,however,itisshownthatchargetransferstatescannotbeproperlydescribedinTDDFT.ForstandardDFTmethods,gradientcorrected,hybrid,etc.thishasbeennoted,andimprovedupon[ 51 ],butstandardmethods,unlikeTDOEPx,stillsuerfromotherlimitations,liketheselfinteractionerrorandtheincorrectlongrangebehaviorofthepotentialsandkernels.Inanapproachthatgivesthe'rightanswerfortherightreason'theseexactconditionsarerequisite.InTDOEPxtherearenosucherrors.Hence,failuresofTDOEPxcomparedtoTDHFhavetobeexclusivelyduetothelocalversusnonlocalexchangeoperator,whichisdemonstrated.Finally,C6coecientsareobtainedforopenshellsystemsfromtheTDOEPxfrequencydependentpolarizability. 82
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(5{3)bhS(1)=bh(1)+bJ(1)+bVX(1) (5{4)bJ(1)=XjZj(2)1 (5{5)=Z(1) (5{6)KS(1)=Xjj(1)j(1) (5{7)wheretheunoccupiedorbitalsareindicatedbyp=a,b,c,d....Theconditionthatthedensity,(1);betheHartreeFockone,isthatfia=hHFijbh+bJbKjHFai=0 (5{8)HF(1)=XjHFj(1)HFj(1) (5{9)which,asiswellknown,iscorrectthroughrstorderincorrelationmeasuredrelativetothesumofHFoneparticleoperators,H0=Pif(i);duetotheMoellerPlessettheorem.ForKSorbitals,fiaisnotzero.Hence,iftheobjectiveweretomaximizethesimilarityintheKSandHFdensity,itwouldthenrequireminimizingaquantitycomposedofKSorbitals,relatedto 13 16 17 48 ].Clearly,thisminimizationwouldhavethetrivialsolutionthathijKjai=hijVXjai. 83
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13 16 17 48 ],sinceitalsotranscendsanyparticularseparationoftheHamiltonianinperturbationtheory.Inparticular,itisnotsubjecttoHKS0=PihS(i);andtherebyalleviatesthepathologicalbehaviorthatgivesnoconvergencewhendeningsuchcorrelationpotentialsfromMBPT(2),forexample[ 13 ].Fortheexchangeonly,however,sucheectsarelessimportant[ 16 17 ].ThereforethispaperwillusethesumofShamHamiltonians,buttheprincipleforderivationremainsthesame.Hence,requiringthat0=(1)KS=hKSjbR0bVjKSi+hKSjbVR0bjKSi (5{12)R0=Xi;ajaii(ia)1haij+XI
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52 ].Inadditiontoitspointwisecharacterrepresentedbythedeltafunction,itdiersfromtheaboveleastsquaresformbythepresenceofthedenominator.Theusualwaytowritetheaboveequationistointroducethenoninteractingdensitydensityresponsefunction, ThisistheusualoptimizedeectivepotentialprocedureofTalmanandShadwick[ 15 22 ],gearedtowardrepresentingtheVXoperatoronanumericalgrid.Theweightfactor,wia(1);isalsointroduced.Notethematrixelementscanbewrittenincongurationspaceas(VX)ai=hajbVXjii 85
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5{18 )thenbecomes, (5{24) Usingweightedleastsquares 5{16 ).Asiscustomaryinpractice,thecomputationalformfortheOEPequationcanbeobtainedbytwiceapplyingtheoneelectronprojector, 53 ]oftheresponsefunctionisthen 5{18 )thatresultsintheYcolumnmatrix. 86
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52 ],unliketheinnitebasisanalogueofX(1,2),whichhastobesingular[ 52 54 ].Thelatterfollowsbecausetheinnitebasiscannotaccountforthearbitraryconstant.However,inanitebasis,XisillconditionedbecausethenullspaceoffunctionsorthogonaltoVXcanalwaysbeaddedtoVX,butwithzeroweight,andthisrequiresthattheOEPequationsproduceacomputed,numericalzero.Thismakestheequationsdiculttoinvert,recommendingasingularvaluedecomposition(SVD)procedure.ThisSVDshouldnotbeconfusedwiththeneedtoexcludelineardependencyinthecomputationalbasis,whichcanalwaysbeachievedindependently;butthat,too,wouldbeaccommodatedinpracticebytheSVDprocedure.TheroleofXistoimposethepointwiseclosenessof^Kand^VX,and,assuch,itiscriticaltothedeterminationofOEPpotentials.IntheabsenceofX,asattemptedbyStaroverov,etal.[ 45 ]unphysicalresultscanbeobtained.Accordingtotheirprocedure[ 45 ],togetHFenergiesanddensitiesfromtheOEPmethod(i.e.tondthetrivialsolutionofEqn( 5{26 )),thenumberofauxiliaryfunctionsusedmustbegreaterorequaltosomenumber,Nmin,whichisthenumberofnonzeroKiamatrixelements.Inthiscaseequation(8)ofref[ 45 ]willhaveoneormoresolutions.Toillustrate,theuseoftheresponsematrixapproach,wereportinFigure 51 theresultsfortheNeatomexchangepotentialsusingthethreebasissetsusedbyStaroverov,etal,plusafourthuncontractedRoosbasisthatweprefer.TheauxiliarybasesarethesameasthemolecularonesandinallcasesthenumberofauxiliaryfunctionsislargerthanthecorrespondingNmin,necessaryforthesolutionofequation(8)ofref[ 45 ].ThetotalenergyvaluesfromTable 51 andcorrespondingpotentialsfromFig 51 arecompletelyoppositetotheresultsofref[ 45 ]becauseofthebrokenpointwiseclosenessandtheabsenceoftheselfconsistentsolution.NotethatoncethepointwiseclosenessisimposedviaX,andaselfconsistentsolutionobtained,thereisnoproblemfortheproperSVDbasedOEPprocedure. 87
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ExchangepotentialsofNeatom,obtainedindierendbasissets Table51. Total(ina.u.)andorbital(ine.v.)energiesofNeatom HOMOBasisNauxNminE(OEP)E(ref[ 45 ])OEPHF AUGCCPVTZ179128.533191128.53327333.5823.16AUGCCPV5Z2515128.545887128.54678614.0123.14AUGCCPV6Z2518128.546287128.54706217.8023.14ROOSADZP6139128.545016128.54659623.7423.14 InspiredbyanobservationofJiang[ 55 ]ofHOMOandLUMOdegeneracyarisinginOEP,Rohretal[ 49 ]oeredaspeciccombinationofmolecularandauxiliarybasis,thatshowedtheireectforHe,thenusingthedirectoptimizationalgorithmproducesdegenerateHOMOandLUMOorbitalenergies.WeperformedOEPcalculationfortheHeatomusingtheoptimizedauxiliarybasisfromTable1ofref[ 49 ],whichcausedtheirdegeneracy,butinsteadofusingthedirectminimizationalgorithmofBroyderFletcherGoldfarbShanno(BFGS)thesolutionoftheOEPequationissolvedbytheinversionofXwiththeSVDprocedure.TheseresultsarepresentedinTable 52 ,demonstratingthat Table52. TotalandorbitalenergiesofHeatom OEPHFOEP2KSOEP2sc Totalenergy,a.u.2.861153652.861153652.905247382.89488588HOMOenergy,e.v.24.97024.97024.52924.787LUMOenergy,e.v.3.59115.5613.6713.618 88
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52 ,eventhenotoriouslyunstablesecondorderOEPprocedure,basedonHKS0hasnoparticulardiculties,either.SincethedirectminimizationprocedureofYangandWu[ 46 ]isbasedonthesameequationastheregularOEPmethod,alloftheproblems,describedinref[ 49 ],arecausedbyusingtheBFGSalgorithminsteadofsolvingtheOEPFredholmequationwiththeSVDprocedurefortheX.Thatmightappeartobesurprisingsinceh=l,wouldmeanthattheresolventoperatorismanifestlysingular,clearlynotthatappropriatetoanondegenerateperturbationtheoryfunctional,whichhasalwaysbeentheintentofsuchorbitaldependentexpressions.However,itisalsoapparentfromthestandard,chainruledierentiationusedbyRohretal[ 49 ],namely 44 ]whomadesuchanapproximationintheGreen'sfunctionwherethedenominatorconsistsofpqbeforesimplifyingthefromtotheoccupiedvirtualseparationinvokedhere[ 16 54 57 ].)SeeGritsenko,etal[ 56 ].Suchaconstantenergydenominatorisarelativelypainlessapproximationhere,sinceavgappearsonbothsidesofeqn.(18),makingitdisappearfromtheequationforVX:Oncethatisdone,aresolutionoftheidentitycanbeinvokedtoeliminatethevirtualorbitalstogive, 89
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59 ]Ofcourse,usingtheusualnitebasissetcomputationaltoolsofquantumchemistry,thedistinctionislessimportant.Thisformulahasrecentlybeenapplied[ 58 ].BecausetheOEPequationisapointwiseidentityhavingtobezeroformallyforallxinanitebasissetitcorrespondstoamanytofewmappinginthegeneralcase.ThedependenceuponthedeltafunctionmakestheOEPproceduresensitivetotheauxiliarybasissetusedinthecalculation.However,theSVDprocedurehandlesthenullspacefunctionsplusanypotentiallineardependencyinthebasissetinafairlyautomaticway[ 60 ].IftheXmatrixisremoved,thisisnotthecase,andsomealgorithmscanresultinunphysicalortrivialsolutionsundercertaincombinationsofauxiliaryandmolecularbases.Choosingtoinvoketheaverageenergydenominatorandtheresolutionoftheidentitybeforetheleastsquaresmminimization,willnotgiveEqn(24).However,onceweeliminatethedeltafunctionthatmakestheapproximationpointwise,wereducetheproblemtotheminimizationofthevariance minVar=minXihij(bK+bVX)2jiiXi;jhijbK+bVXjjihjjbK+bVXjii(5{34)thatcanbeusedtodeneaVX,butonethatisnotpointwise,and,consequently,doesnotsatisfyalltheconditionsabove.ThisfurthersimplicationoftheweightedleastsquaresapproachinEqn( 5{26 )hasalsobeenconsideredrecently[ 58 ].Ultimately,oneprimarycriteriaforthebestexchangeonlypotentialshouldbethesatisfactionoftheJanaktheorem.AsshowninTable 51 ,andpointedoutpreviously[ 16 48 ]thisisdiculttoachieveinanynormalbasisset. 90
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5.2.1TheoryandImplementationWithintheadiabaticapproximationtheTDOEPequationshavethefollowingmatrixform( 5{35 ) 5{36 ) where with (g)=2Xi;a;b(Kba+VOEPba) ("i"a)("i"b)+2Xi;j;a(Kia+VOEPia)(+) ("i"a)("j"a)2Xi;j;a;b(+) (5{39) Oncetheeigenvalueproblem( 5{35 )issolved,wehavethesetofexcitationenergies,!. 91
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5{40 )shouldbesolved 5{40 )shouldbesolvedfortheimaginaryfrequenciesi!fromwhichthecoecientscanbycalculatedby 3(xx(i!)+yy(i!)+zz(i!)).Integrationofequation( 5{43 )wascarriedoutbyGaussChebyshevquadrature.Toensurethecorrect1 30 ]wasused.Asaseedpotential,weusedtheoneproposedbySlater[ 38 ] 36 ].Forallcalculationstheuncontracted 92
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27 ]hasbeenused.ResultsforexcitationenergiesarepresentedinTable 53 Table53. Excitationenergies(Vvalencestate,RRydberg) TDHF"OEPi"OEPaTDOEPSVWNCCSDExp CN2(V)4.188.164.911.961.521.322+(V)5.4710.585.483.193.623.22CO+2(V)7.559.988.233.033.433.262+(V)11.149.3710.824.996.145.82CH32A01(R)6.526.536.425.005.885.722A002(R)7.937.977.935.937.187.44N4P(V)9.8411.119.8810.9810.8410.354P(R)13.5913.1613.1811.76 AsfollowsfromthersttwocolumnsofTable 53 ,TDHFandTDOEPxproduceresultsofapproximatelythesameaccuracy,aswouldbeexpectedbythefactthatthelocalexchangeinTDOEPismeanttobeakindofleastsquaresttotheHFnonlocalexchangepotential.Thiscertainlymakestheoccupiedorbitalsquitesimilar,butthespectrumofunoccupiedorbitaleigenvaluesisverydierent,asshowninTable 54 ,sinceTDOEPxwillgenerateaRydbergtypeseriesinsteadofanythingliketheHFvirtuals.Asiswellknown,thelatteraredeterminedinapotentialofnelectrons,makingthemappropriateforelectronattachedstates,whiletheoccupiedonesfeelapotentialofn1electrons.Tothecontrary,theorbitalsobtainedinOEPxhavethesamepotentialforanelectronintheoccupiedandunoccupiedorbitals,whichiswhythelattermorenearlysimulateRydbergstates,assomeoftheunoccupiedorbitalswillhavenegativeorbitalenergies.Ofcourse,inanitebasisaslongasthespaceseparatelyspannedbytheoccupiedandtheunoccupiedorbitalsisthesame,therewouldbenodierenceintheresults,thoughthediagonalvalues(butnotthetrace)wouldchange.Thesignicant 93
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Orbitalenergies(ine.v.)ofNeatom OEPxHFEOMCCSD 1s839.67891.78871.112s47.3652.5248.112p23.7423.1421.283s(3p)5.213.663.703p(3s)3.113.943.624p5.6616.0815.66 discrepancieswiththeexperimentaldatacanbeexplainedbythefactthatelectroncorrelationeectsarenotintroducedbyeitherTDHForTDOEP.Togetsomemeasureofthecorrelationeect,coupledcluster(CC)resultsfromEOMCCSDarereported[ 61 ],andtheSlaterVoskoWilkNusiar(SWVN)functionalisalsoused.Thelatterproducesgoodresultsforvalencestates,buttheenergiesoftheRydbergstatesareunderestimated.ThisunderestimationisaconsequenceoftheincorrectasymptoticbehavioroftheSWVNpotential,forwhichJanak'stheoremisnotsatised(seeTable 55 )andisawellknowncharacteristicthatstandardexchangecorrelationpotentialsdonotoeraKoopmanstypeapproximationfortheorbitalenergies.HoweverforOEPexchangeonlyandwithcorrelation,ithasbeenproventhattheorbitalenergiesoerameaningfulKoopmansapproximationtoprincipalIP'sforthevalenceandmidvalencestates[ 16 ],soRydbergstatesshouldpotentiallybebetterdescribed.Ofcourse,denitivecomparisonwouldrequireOEPwithcorrelationaspresentedelsewhere[ 62 ],andthisisseenintheOEP2scorbitalenergies[ 16 48 ],butafullTDOEPtreatmentofexcitedstates[ 62 ]requiresthecorrelationkernel.Thisdoesnotappeartobeaviablerouteyet. Table55. Ionizationenergies(ine.v.) HOMOEHFOEPSVWNHFOEPSVWNExp CN14.1614.619.7916.2216.3214.8013.60CH310.4611.195.388.988.9910.099.84N15.5316.148.4113.8913.8915.0014.53 ResultsforthepolarizabilitiesandC6coecientsarepresentedinTables 56 and 57 .Inallcases,thestaticpolarizabilitiesobtainedwithTDOEPareveryclosetothe 94
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63 ]andDreuwetal[ 51 ].Accordingtothelatteranalysis[ 51 ]eveniftheexactexchangecorrelationfunctionalwereknown,itwouldstillbeimpossibletogetaproperdescriptionofchargetransferexcitationswithinTDDFT.ConsidertheexampleofHe...Be.Thechargetransfer1statecalculatedwithTDHFandTDOEP,takenwiththeTammDanco(monoexcitedCI)approximationforsimplicity.ForbothatomstheuncontractedROOSADZPbasissetwasused.Potentialcurvesofthe1chargetransferexcitedstatearepresentedonFig 52 .Thepotentialcurve,calculatedwiththecongurationinteractionsinglesmethodexhibitsthecorrect1 5{36 )areequaltozeroandtheexcitationfrequencyisequaltothedierenceoftheLUMOorbitalenergyoftheBeatomandtheHOMOenergyoftheHeatom.WhatwasnotaddressednumericallyinDreuwetal[ 51 ]wastheeectofhavingaproperselfinteractionandasymptoticbehaviorinDFT,whichcanonlybeachieved 95
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A1chargetransferedexcitedstateofHe...Be withOEPmethods.WereportthoseinFig 53 .Obviously,theorbitalenergydierence Figure53. LUMOHOMOorbitalenergydierence behavesverydierentlyforOEPandHFsinceinbothcasestheHOMOorbitalenergycorrespondstoanionizationpotentialapproximation,buttheLUMOenergycorrespondstoaRydbergstateapproximationfortheOEPmethodandtoanapproximateelectron 96
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5{36 )ascorrections,itisapparentthatTDDFThasaqualitativelywrongbehaviorineachorderwhendealingwithchargetransferexcitations.IfonearguesthattheoccupiedorbitalsintheHFandOEPcalculationsarenearlythesame,andtheyspantheoccupiedspace,thentheunoccupiedspacewouldbethesame,too.However,evenifthespaceswerethesame,thedierencebetweenthenonlocal^Kandlocal^Vxissucienttokeepthelatterlocalexchangefromeverbeingabletodescribeanelectronattachedstate.Thus,itisnotpossibletoovercomethelocaloperator'sfailureeveninthefullspaceforthecalculations. 16 52 ],anddoesrequiretheuseofageneralized(SVD)inverseortheequivalent.ItwasalsoshownthatundertherightconditionstheexactHFresultcanbeobtainedbyanOEPlikeprocedureasatrivialsolutionofaweightedleastsquaresprocedure.Anypointwiseconditionhastobeamanytofewmappinginanitebasisset.TheeliminationofdenominatorsfromworkingexpressionsforOEPx[ 20 44 56 ]canbedoneasshown,toobtainthecomputationaladvantageofusingonlytheoccupiedorbitalsintheOEPprocedure.Thedirectoptimizationalgorithmasimplemented,however,hasledtowork[ 45 49 ]thathassuggestedincorrectconclusions.TheTDOEPmethodwasimplementedforspinpolarizedopenshellsystems.ResultsforseveralopenshellsystemsshowverysimilartimedependentbehaviorforthetimedependentresponseoftheOEPandtheHFpotentials.Thecorrectlongrange 97
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64 { 66 ]TheproblemofchargetransferexcitedstatesinTDDFTrelatestotheinabilityofthezerothorderorbitalenergyapproximations,astheHFvirtualorbitalsandTDOEPexcitedorbitalsshowqualitativelydierentbehavior.IftheHFandOEPoperatorswerethesame,apropertreatmentusingthewholespaceshouldbeabletoovercomethislimitation,eveniftheusualzerothorderapproximationoeredbyTDDFTwouldbeapoorone.However,thisdierencearisesfromtheformoftheoperator,localversusnonlocal,and,assuch,remainsafundamentalproblemfortheDFTmethoditself,ashasbeenobserved[ 51 ]Inconclusion,forallthepropertiesconsideredinthepaper,exceptforchargetransferexcitedstates,thereseemstobenosignicantdierencebetweentheresultsofTDHFandexchangeonlyTDOEP,asonemightexpect.Hence,thefocusshouldremainonthecorrelationpotentialasitisinabinitiodft[ 16 17 48 62 ]. 98
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Staticpolarizabilities(ina.u.) TDHFTDOEPSVWNCCSD CNxx11.9811.9115.6312.50yy11.9811.9115.6312.50zz18.5518.4125.0023.4214.1714.0818.7516.166.576.509.3810.94CO+xx6.666.669.389.38yy6.666.669.389.38zz12.2212.1514.0612.508.528.4910.9410.425.545.504.693.13CH3xx15.0615.0018.7515.63yy14.0413.9818.7515.63zz15.0615.0018.7515.6314.7214.6618.7515.631.021.020.000.00Nzz6.246.236.255.94 Table57. IsotropicC6coecients(ina.u.) TDHFTDOEP CN...CN93.9792.06CH3...CH394.2793.40N...N19.2819.21 99
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20 ]andimplementedformoleculesbyHirataet.al[ 21 50 ].NumericalresultsinthelatterpapersshowareasonabledescriptionofbothvalenceandRydbergexcitedstates.Reasonablestaticanddynamicpolarizabilitiesarealsoobtained.GoodresultsofTDOEPxforexcitedstatesandpropertiesareobtainedpartlyduetothecorrectasymptoticbehavioroftheexchangepotential,butalsoduetotheeliminationoftheselfinteractionerror.Inparticular,OEPTDDFTxtendstobesuperiortostandardapproacheslikeTDDFTbaseduponthelocaldensityapproximation(LDA)orBeckeLeeYangParr(BLYP)functionals.Similarly,exchangeonlyOEPwithexactlocalexchange(EXX)[ 22 23 ]hasbeenshowntogreatlyimprovebandgapsinpolymers[ 23 ]AnotheradvantageofOEPbasedmethodsisthatsincevirtualorbitalsintheexchangeonlyDFTaswellastheoccupiedorbitalsaregeneratedbyalocalpotential,whichcorrespondstotheN1particlesystemduetothesatisfactionoftheselfinteractioncancellation,thedierencesbetweenorbitalenergiesofvirtualandoccupiedorbitalsoeragoodzerothorderapproximationtotheexcitationenergies[ 16 17 ].ThisisnotpossibleinthecaseofHartreeFocktheory,withoutaddingaVN1potential[ 67 68 ],sinceoccupiedorbitalsaregeneratedbyanN1particlepotentialandtheenergiesofunoccupiedorbitalscomefromNelectronpotential,andthereby,approximateelectronanities.InKohnShamDFT
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^Vxc(r)=Exc 21 50 ].However,onceourobjectiveisthirdordermolecularproperty,theessentialnewelementinTDDFTisthesecondexchangecorrelationkernel,whichinanadiabaticapproximationisdenedasthesecondfunctionalderivativeoftheexchangecorrelationpotentialwithrespecttothedensity 37 62 ].InthispaperweusethatformalismandapplyittotheexchangesecondkernelforOEPxbasedtimedependentDFT.Afterdiscussingsomepropertiesofthesecondkernel,wereportnumericalresultstoobtainhyperpolarizabilities,comparedtothosefromHartreeFockandcoupledclustersinglesanddoubles(CCSD).Thereisalonghistoryofthetreatmentofhyperpolarizabilitiesandassociatednonlinearoptical(NLO)properties.See[ 69 ]forareview.Inparticular,issuesoftheincorrect 101
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64 65 ]pointstothenecessityofanimprovedtheory.Arigorous(abinitio)DFTanaloguestartswithexchangeonly,TDOEPx. 6.1.1TimeDependentDensityFunctionalTheoryResponsePropertiesWhenanexternaltimedependentelectriceldisapplied,theperturbationcanbewrittenintheform( 6{2 ) (A+B)U=h(6{3)ThematricesA,Bhavethestructure,givenbyEqn( 5{36 )andhis 70 ] 102
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6{6 ) Fpq=hpq+Xa;i(Apqai+Bpqai)Uai ij=Fij 6{8 ) (6{9) Takingintoaccountthefactthat (6{11) where 103
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6{11 )canbewrittenas (6{13) andP1(r2;r6;r7)canbepresentedasasumofdiagrams(F1)(F6)of( 6{14 ) (6{14)Inthesamewaythethirdtermofeq(16)canberewrittenas (6{15) whereP2(r1;r6;r7)givesthesumofdiagrams(F7)(F12)of( 6{16 ) (6{16)ThediagrammaticexpressionforQ(r6;r7)containsterms(V1)(V6)and(FX1)(FX8)of( 6{17 )and( 6{18 )[ 62 ] (6{17) 104
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Functionalderivativesfromtheset(V1)(V6)canbepresentedbytwotypesofdiagrams.Thersttypecontainstheexchangekernel,diagrams(F13)(F18)of( 6{19 ) (6{19)Thesecondtypecontainstheexchangepotential,givingdiagrams(VX1)(VX8)of( 6{20 ) (6{20) Thenotationr4nr5nr6meansthatforeachparticulardiagramallsixpermutationsofr4,r5andr6shouldbetaken.Thefunctionalderivativesfromdiagrams(FX1)(FX8)leadtotheset(SX1)(SX14)of( 6{21 )and( 6{22 ). (6{21) 105
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InterpretationofalldiagramswhichcontributesintothesecondkernelanddetailsoftheimplementationaregiveninAppendix B 38 ] 6{20 )willcanceldiagrams(SX7),(SX8)and(SX11)(SX14)of( 6{21 )and( 6{22 )becauseoftheHOMOconditionandthebehavioroftheHartreeFockexchangeforthecaseoftwoelectrons.Diagrams(VX6)and(VX7)willcanceldiagrams(SX1)and(SX2)becauseofEqn( 6{23 ).Diagrams(SX5),(SX6),(SX9)and(SX10)willcanceldiagrams(F1)(F4),(F7)(F10)and(F13)(F16)becauseofEqn( 6{24 ).Finallythesumofthediagrams(SX3)and(SX4)cancelthesumof(F5),(F6),(F11),(F12),(F17)and(F18).Thusforthespecialcaseoftwoelectrons 106
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6{25 )mightbeusedasahintfordevelopingnewdensityfunctionals,asnoneoftheexistingfunctionalsexhibitsuchbehavior. 27 ]areusedandequlibriumgeometriesaretakenfromref[ 36 ].TheresultscalculatedwithdierentmethodsarepresentedinTable 61 .Intherstfourcolumnsarestatichyperpolarizabilitiesobtained Table61. Hyperpolarizabilitiesofseveralmolecules(ina.u.) HFOEPHFOEPxLDACCSD LiHzzz312.130314.910312.126621.093691.406xxz201.150207.868201.641404.297204.171COzzz31.01624.86429.45566.40627.343xxz3.0734.6603.11813.2035.859CNzzz1.2990.2113.9061.953164.02xxz17.65115.64618.12162.50068.359H2Ozzz6.7136.2397.81213.6725.859xxz0.4970.6041.2156.1383.870yyz10.86610.3211.84723.4377.645 withHartreFocknonlocalexchangeandwiththetwolocalexchangeoperators:OEPHF,whichmeansequation( 6{5 )withoutthesecondkernelterm,andOEPx,whichmeansalltermsinequation( 6{5 ).ThehyperpolarizabilityvaluescalculatedwiththeOEPxmethodareingoodagreementwiththecorrespondingHartreeFockvalues,andgenerallynobetter.ThisgoodagreementcanbeexplainedbythefactthatOEPxmethodisfreefromtheselfinteractionerror,hasthecorrectlongrangeasymptoticbehavior,andthecorrectHOMOvalues.AshyperpolarizabilitiesareresponsepropertiesofthirdordertheircalculationwithintheDFTmethodrequiresverypreciseexchangecorrelationpotentials.OnlypotentialswhichsimultaneouslysatisfyalltheoremsandconditionspertainingtotheDFTexchangepotentialscanbeexpectedtoreproduceeventheHartreeFock 107
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66 ]itwaspointedoutthatnoneoftheexistinghybridandasymptoticallycorrectedfunctionalsarecapableofreproducingHartreeFockhyperpolarizabilities.Thesemiempiricalmethod,developedthere[ 66 ]istheonlyknownwaytogetreasonablehyperpolarizabilitieswithconventionalfunctionals.However,theOEPxpotentialsandkernelsaretheonlyrigorous,purelyabinitiowaytoobtainhighorderproperties,withinasolelyDFTframeworkoflocalpotentialsandkernels.ForthecalculationofhyperpolarizabilitieswiththeHartreeFockmethod[ 71 ]thelasttermineq( 6{5 )shouldbedropped.ItsimportancefortheDFTschemecanbeestimatedfromthesecondcolumnofTable 61 ,whereOEPhyperpolarizabilitieswerecalculatedwithoutthelasttermofequation( 6{5 ).FortheexampleoftheLiHmoleculetherelativecontributionofthelasttermissmall.However,goingfromclosedshelltoanopenshell,likefortheCNmolecule,therelativecontributionofthethirdkernelbecomesmuchgreaterand,sometimes,canevenchangethesignofhyperpolarizabilities.ThedierencesbetweentheOEPxandthecoupledclustersinglesanddoubles(CCSD)methodarebecauseintheexchangeonlyOEPmethodelectroncorrelationhasnotbeentakenintoaccount.Forthecalculationofhighorderproperties,whichcanbecomparedwithexperimentaldata,accountingfortheelectroncorrelationiscritical[ 69 72 73 ],andgenerallyCCSDisfairlygood,thoughaddingtriplesoersomeimprovement.Nevertheless,thepresentresultsshowwhatkindofpropertiesexchangepotentialsandkernelsmusthavetobeabletodescribehigherorderpropertieswithintheframeworkofDFT,andasmentionedpreviously,noneofthecurrentsetoffunctionalsforstandardDFTshowsthisbehavior. 108
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16 ]IthasbeenshownthattherigorousexchangetreatmentinTDOEPandTDHFprovidehyperpolarizabilitiesthatcloselyagreewitheachother,asonewouldexpect,althoughthiswillnothappenforchargetransferexcitedstates[ 51 74 ].ThisclearagreementwithTDHFisincontrasttoconventionalexchangefunctionalsthatseverelyoverestimatehyperpolarizabilities.OEPpotentialsandkernelsarenotonlycapableofdescribingexcitationenergies,butalsohigherorderproperties.SincenoneofthestandardDFTfunctionalsshowsthecorrectanalyticpropertiesofthesecondkernel,theOEPformulasandresultscanbeusedforthetestingandcalibrationofnewdensityfunctionals. 109
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(A{1) where "i"a)=Zdr'p(r)'q(r)(r) (A{2) (VC11)+(VC22)=2Xa;b;c;i;j;k(+) ("i+"j"a"b)("i"c)("k"b) (A{3) (VC12)=2Xa;b;c;d;i;j (A{4) (VC13)+(VC31)=2Xa;b;c;i;j;k(+) ("i+"j"a"b)("i+"k"a"b)("i"c) (A{5) 110
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("i+"j"a"b)("i+"j"a"d)("i"c) (A{6) (VC15)=2Xa;b;c;d;i;j (A{7) (VC16)+(VC25)=2Xa;b;c;i;j;k(+) ("i+"j"a"b)("i"c)("k"b) (A{8) (VC17)=4Xa;b;c;d;i;j (A{9) (VC18)+(VC28)=4Xa;b;c;i;j;k(+) ("i+"j"a"b)("i"c)("k"a) (A{10) (VC19)+(VC110)=4Xa;b;c;i;j;k(+) ("i+"j"a"b)("i"c)("k"c) (A{11) (VC111)+(VC37)=4Xa;b;c;i;j;k(+) ("i+"j"a"b)("k+"j"a"b)("k"c) (A{12) (VC112)+(VC48)=4Xa;b;c;d;i;j(+) ("i+"j"a"b)("i+"j"d"b)("i"c) (A{13) 111
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("i+"j"a"b)("i"d)("i"c) (A{14) (VC115)+(VC215)=4Xa;b;c;i;j;k(+) ("i+"j"a"b)("i"c)("k"a) (A{15) (VC116)=4Xa;b;c;i;j;k (A{16) (VC21)=2Xa;b;i;j;k;l( (A{17) (VC23)+(VC32)=2Xa;b;i;j;k;l(+) ("i+"j"a"b)("i+"l"a"b)("k"a) (A{18) (VC24)+(VC42)=2Xa;b;c;i;j;k(+) ("i+"j"a"b)("i+"j"a"c)("k"a) (A{19) (VC26)=2Xa;b;i;j;k;l( (A{20) (VC27)=4Xa;b;i;j;k;l( (A{21) 112
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("i+"j"a"b)("k"c)("k"a) (A{22) (VC211)+(VC47)=4Xa;b;c;i;j;k(+) ("i+"j"a"b)("i+"j"b"c)("k"c) (A{23) (VC212)+(VC38)=4Xa;b;i;j;k;l(+) ("i+"j"a"b)("l+"j"a"b)("k"a) (A{24) (VC213)+(VC214)=4Xa;b;i;j;k;l(+) ("i+"j"a"b)("k"a)("l"a) (A{25) (VC216)=4Xa;b;c;i;j;k (A{26) (VC33)+(VC35)=Xa;b;i;j;k;l(+) ("i+"j"a"b)("i+"k"a"b)("l+"j"a"b) (A{27) (VC34)+(VC45)=Xa;b;c;i;j;k(+) ("i+"j"a"b)("i+"k"a"b)("i+"j"b"c) (A{28) (VC36)+(VC43)=Xa;b;c;i;j;k(+) ("i+"k"a"b)("i+"k"c"b)("i+"j"c"b) (A{29) 113
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("i+"k"a"b)("i+"l"a"b)("i+"j"a"b) (A{30) (VC311)+(VC412)=2Xa;b;c;i;j;k(+) ("i+"k"a"b)("i+"j"a"c)("i+"j"a"b) (A{31) (VC312)+(VC411)=2Xa;b;c;i;j;k(+) ("i+"k"a"b)("i+"k"a"c)("i+"j"a"c) (A{32) (VC313)+(VC314)=4Xa;b;c;i;j;k (A{33) (VC44)+(VC46)=Xa;b;c;d;i;j(+) ("i+"j"a"c)("i+"j"a"b)("i+"j"d"b) (A{34) (VC49)+(VC410)=2Xa;b;c;d;i;j(+) ("i+"j"a"c)("i+"j"a"b)("i+"j"a"d) (A{35) (VC413)+(VC414)=4Xa;b;c;i;j;k (A{36) (VC51)+(VC61)=2Xa;b;c;i;j;k(+) ("i+"j"a"b)("i"c)("k"a) (A{37) 114
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(A{38) (VC53)+(VC81)=2Xa;b;c;i;j;k(+) ("i+"j"a"b)("i+"k"a"b)("i"c) (A{39) (VC54)+(VC712)=2Xa;b;c;d;i;j(+) ("i+"j"d"b)("i+"j"a"b)("i"c) (A{40) (VC55)+(VC78)=2Xa;b;c;d;i;j(+) ("i+"j"a"b)("i+"j"a"d)("i"c) (A{41) (VC56)+(VC68)=2Xa;b;c;i;j;k(+) ("i+"j"a"b)("i"c)("k"b) (A{42) (VC57)=2Xa;b;c;d;i;j (A{43) (VC58)+(VC66)=2Xa;b;c;i;j;k(+) ("i+"j"a"b)("i"c)("k"a) (A{44) (VC59)=2Xa;b;c;d;i;j  