A EFCJECTEC IIAEILTCNIAN APPROACH
TO PCLYATCMIC SYSITES
Ey
JACK A. SMITH
A DISSERTPTICN PRESENTED TO THF GRADUATE COUhCIL OF
THE UNIVERSITY OF FLCRIDA
IN PARTIAL FULFILLFFNT OF THE BEQUIREFNTS FCf THF
EFGREF CF ECCTCR OF FIHILOSOPFY
UNIVERSITY OF FICRIDA
1978
to
Ndlis~a
ACKNOWLEDGEMENTS
I would like to acknowledge the guidance and assistance
given by Frofesscr Yngve Ohrr in the course of this work. I
also wish to take this opportunity to thank all the members
of the Cuantum Theory Froject for such a stimulating
environmErt in which tc work and study. I owe special
thanks to Prcfessor PerOlov L6wdin for providing me the
many cppcrtunities to attend summer schools, winter schools,
symposia end tte like in such places as Sweden, Ncrway,
Belgium, Sanitel Island, Palm Ccast and last, but not least,
Gainesville.
I also ackrowledre the financial support of the Northeast
Reqicnal Eata Cen4er, the Air Force Office of Scientific
Research (AFCSF 742656C), and the National Science
Foundaticr (NSF CHE7401948, SMI7623036).
iii
1
TABLE OF CCNTFNTS
ACKNCWLEEGEMENTS ...................................... iii
LIST OF TABLES ........................................ V
LIST OF FIGURES ....................................... vi
ABSTRACT .............................................. vii
CHAPTER
I. INTRCDUCTION ................................. 1
II. THE EFFECTIVE CNFELECTFCN CPEBATOF
AND THE PARTITIONING OF ITS EIGENSFACE ....... 5
III. THE ENEFGYDFFENCFNT PSEUFOPOTENTIAL ......... 19
IV. A MOEEL APPRCXIMATICN TC THE EFFECTIVE
CNEELECTECN INTEFATCMIC CCULCMB POTENTIAL ... 28
V. AN ATPRCXIMATE NCNLOCAL INTERATCIIC
EXCHANGE FCTENTIAL ........................... 37
VI. A MODEL HAMILTCNIAN AND AN APERCXTIATE
ELECTRCO FSOEAGATOR .............. .............. 39
VII. THF GRANDCANCNICAL HARTREEFCCK SCHEME ...... 45
VIII. A SEIFCCNSISTENT CHANGE AND
CONFIGOBATION FROCEDURE ...................... 50
TX. APFLICATICNS ........ ... ..... ..... ............. 58
A. Nitric Oxide Mclecule .................. 58
B. Nickel (100) Surface .................. 78
C. Nitric Oxide cn Nickel(100) Surface .... 8
X. EISCUSSION ................................... 107
BIEIICGEAHY ........................................... 111
EICG nAPHiCAL SKETCH ......... .......... ................. 115
iv 
LTST OFP ABTES
1. Basis sets for nitrogen and oxyqon.................. 62
2. Model potential exFcnents for ritrcgen and oxygen... 62
3. Calculated and experimental ionization potentials
for nitric cxide (X2n)...... ........................ 66
4. Calculated and experirertal spectroscopic constants
fcr ritric oxide (X2n) ............................... 77
5. Basis set for nickel............... ................. 83
6. Icdel potential exponents for nickel................ 83
7. Calculated ard experimental ccre(lI) binding
energies for nitric oxide on nickel (100) surface.... 102
8. Calculated spectrosccpic constants for nitric oxide
cn nickel (10C) surface............................... 105
LIST OF FIGURES
1. Electronic and nuclear coordinates for the diatomic
molecule A F ....................................... 6
2. Calculated and experimental photoelectrcn spectra
for nitric oxide (X2n) ............. ................ 66
3. "Hiddenline" plots of the atomic crbitals cr the
nitrccen atc ......................................... 71
4. "Hiddenlinp" plots of the atomic crbitals on the
oxyqer atcm..... ......... ......................... 73
5. Calculated and experimental (Mcrse) potential enercy
curves for nitric oxide (X2T) ....... ................ 76
6. Cluster models for the bulk, surface, and fourfold
hole adsorption site................................ 82
7. Calculated pctoelectrcn spectra of nickel(atom),
nickel(metal), and nickel(surface) .................. 85
8. Total density contours for nitric oxide (top) and
nitric oxide on nickel(1nn) surface (bottom) ........ 93
9. Siqra density contours for nitric oxide (top) and
nitric oxide on nickel (100) surface (bottom) ....... 95
10. Pi dersity ccntcurs for ritric cxide (tcp) and
nitric oxide on nidkel(100) surface (bottom)......... 97
11. Calculated photceletrcn spectra of nitric oxide
on nickel (1CC) surface...... ......................... 100
12. Calculated potential energy curves for nitric oxide
(,) ard nitric oxide cn nickel(100) surface (*)..... 104
vi
Abstract of Dissertation Presented to the Graduate
Council cf the University of Florida in Partial Fulfillment
of the Bequirements fcr the Deqree of Doctor of Philosophy
A PPOJ'CTED HAFIIlCNIAN APERCACH
TC ECLYATCMIC SYSTEMS
by
Jack A. Smith
June 1978
Chairnar: Yncve Ohrn
Major Department: Chemistry
A method of treating polyatcmic systems, finite or
extended, is presented whichh fully exploits their "atcmin
molecule" nature. Within an independentparticle model a
partitioninq technique is applied to +he projection of the
full pclyatomic space into many atomic suhspaces. The
sutspaces are then each coupled tc one another approximately
thrcuqh secondorder in overlap in a piecewise self
consistent fashion. The inherent localized picture allows
for an approximate fcrm of the interatomic interactions
without resortinq to neqlect of any differential overlap or
use of any empirical parameters. Molecular orhitals and
energies may he obtained from an approximate electron
propaqatcr which is tasted on a ncdel Hamiltonian built up
vii
frcr atcric oneelectron Hamiltcrians in iliagonal form.
Symmetric orthonormalization of these orbitals gives a
density aatrix whichh can be used as a guide for charge
redistribution within the system. A generalization cf the
HartreeFcck apFroximaticn based on a statistical ensemble
is employed which permits the use of fractional occupation
numbers in the atomic configurations. Application cf the
method here includes the nitric cxide molecule and its
chemisortrion on a nickel(100) surface.
viii
CHAPTER I
INTRODUCTION
In the Quantun mechanical treatment of polyatomic
systems, whether they he finite cr extended, one feature
consistently emerges, and that is they appear to be built up
from "atcms". The reasonable success of the cellular
(Slater, 1934), valencebond (VE) (Gerratt, 197U), atomin
molecule (AIM) (joffitt, 1951), and similar buildingblock
(Adams, 1971; ilbert, 1972) approaches is indetted tc this
feature. However, as cne prcqresses from atoms to molecules
serious complications are encountered in the numerical and
analytic techniques which have been so successfully applied
tc atoms. These complications are largely due to the less
of spherical symmetry and onecenter expansions. For
numerical techniques, the threat of multidimensicral
inteqraticns and quadratures has led tc cellular
approximations tc the potential (Slater, 1934) and local
approximations fc the exchange interactions (Slater, 1971).
Cn the cther hand, analytic methods are faced with the
tremendous number cf complicated nulticenter tuoelectron
inteqrals, and one is usually forced to sacrifice both
quality and quantity in the selection of basis sets (Dunning
and Hay, 1977). The exclusion of core electrons by means of
PFfective potentials (Kahn et al.,1S76) has become a popular
means of reducing 'he number of integrals. A whcle gaaut of
serierpirical mthods have also been devised and constantly
revised tc help cverccfe these difficulties.
One wonders, though, whether cr not the ability to even
describe the "atoms" within a system is lost in all these
sacrifices. This brings us tc the theme of the present
work. ie wish to maintain as much as possible a complete
and accurate description of the "atoms" in a system at the
controlled expense of an approximate, but sufficiently
rigorous, ncrempirical treatment of the interatomic
interactions. Some corollaries to this theme will be to
keep, intact, the molecular Ilamiltonian while partitioning
the discrete eicenspace (the finite space spanned by the
discrete eiqenfunctions) of an effective cneelectron
operator; tc onplcy analytic techniques enabling proper
treatment of the exchange interacticns; to use suitable
basis sets, such as Slatertype orbitals (STO's) of at least
"doublezeta" (E2) quality with polarization functions (DZP)
when needed; to correctly compute and retain all onecenter
inteqrals and all twocenter oneelectron integrals; to
reduce all multicenter twoelectrcn integrals to twocenter
oneelectron integrals by exploiting the !ccali2ed picture,
but %ithcut resorting to empiricism or neglect of any
differential overlap; and to establish atcmic valency as a
tetterdefinec ard wcrkatle ccncep*.
In the next chapter the effective oneelectron
Hamiltonian operator and the partitioning of its eigenspace
will te develcped. Chapter III will deal more explicitly
%ith the pseudopotential which is derived frfo the
partiticrinq in chapter II. 'he approximate forms of the
interatomic Cculcmtic and exchange interactions are given in
chapters IV ard V. Chapter VI is devoted to the piecewise
construction cf a model molecular oneelectron Hamiltonian
which when used in the moment expansion of the electron
crcpaqatcr gives rise tc a Cmdel Fcck matrix as its first
icient. Solution of the corresponding eigenvalue problem is
discussed together with an appropriate population analysis
for redistributing electronic charge. In chapter VI a more
general form of the HartreeFock approximation is described
which allows for the definiticr of a selfconsistent
potential from a density ccrrespcnding to a statistical
ensemble with specified occupation numbers. The prchlem of
spinorbital cccupaticn assignments is the topic of chapter
VIII. In chapter IX the method is applied to nitric oxide
and its chemisorption cr a nickel(100) surface. Separate
treatments for the nitric oxide molecule, the nickel netal,
4
and the nickel (CO) surface are included to emphasize the
applicability cf the method to various systems. The final
chapter is a general discussion of some pros and cons cf the
metlcd ard score ideas for future icrk to alleviate any
shcrtcominqs the method miqht have.
CHAPTER II
THE EFFECTIVE CNEFIFCTRCN CPERATCR ANE
'HE PAPTTTICNING OF ITS EIGENSFACF
In order tc illustrate the partitioning of a polyatomic
system into many atomic subspaces it is sufficient, at least
for the ncment, to consider a diatcmic molecule, AB. In
much of ktat is about tc fcllcw, specific attention will he
focused on the case of a diatomic acleclee. This is dcne
strictly for clarity, and generalization to polyatomic
systems shall be made at various times. The method is by no
means restricted tc diatomic molecules, even though its
generalization might not appear obvious at times. In
connecticr with this, we will mostly be concentrating on a
single atcm (A) at any one time, while the remaining atcms
(B) will be considered as its environment. We shall usually
denote this by tie use of suFerscrirts A or B, respectively;
but when no specific reference is made, atom A is tc be
assumed. For cases other than diatomics, primes will also
be used. See Figure 1 for other notation conventions tc be
adhered tc throughout this and remaining chapters.
e2 r1B
R r1
/>RB
Figure 1. Electronic and nuclear coordinates for the
diatcmic molecule AB.
The average value of the ncnrelativistic manyelectron
Paniltonian, with resFect to a proper choice of density
operator (see chapters VT and VII), for the diatomic
molecule can te written in second quantization as
(A) A + (A) AA AA + +
= h b + 1/2 ( (J K )
i i i i ij i i i j j i
(B) + (B) BP BB + +
+ B h + 1/2 (J K )
i i i i ij ii ii i j 1 i
(A) (B) AE kB + +
S1/2 ( K )
i ii ii i j j i
(B) (A) FA BA + +
+ 1/2 (J K ) (IT1)
i i ii ii i i i i
where the a's and t's are field operators corresponding tc
spinortitals assigned tc ators A and B, respectively, and
the summations have been restricted accordingly. let us
assume that the spinorbitals assigned to ator A are
expanded in a tasis centered cr A and the spinortitals
assigned tc atom B are likewise centered on B. Such a total
energy functional leads to the effective oneelectron
operator
2 A 1 A A
F = 1/2 0 Z r + J K
1 1 1A 1 1
B 1 B B
Z r + J K (II2)
1E 1 1
where the Coulcrb and exchange operators associated with
center A are defined by
A (A) A A 1 A
J q (1) = q (1) (113)
1 i i i 12 i
and
A (A) A A 1 A
K 4(1) = q <4 Ir It> (1) (114)
1 i i i 12 i
where the Dirac brackets here denote integration over the
coordinates of electron 2 and summation over its spin. The
q's are the sFincrtital occupation numbers. Analogous
definiticrs for the Coulomb and exchange operators on atcir
are assumed. The sFircrbitals are no restricted to be
eioensoluticns c a pseudoeigenvalue problem,
F 4 (1) I S (1), (115)
althcuqh they lust be solved for selfconsistently since the
operator deperds on their thrcuqh the Coulomb and exchange
oTerators.
At this point we have dore nothing but invoke the
HartreeFcck approximation with the only twist being the
localized restriction on the spincrbitals. The canonical
HartreeFcck scheme would require that the spinortitals be
symmetryadapted, that is, invariant (except possibly a
change in sign) unler the operations of the molecular
symmetry point qroup. This, of course, is an unnecessary,
but convenient, restriction. Only the total state need be
invariant, and such invariance can be achieved with
equivalent ortitals (Hurley et al, 19c3), which transform
into each ctFer under the operations of the group. Such
ortitals are more general and can usually be welllocalized.
We shall use such localized equivalent orbitals in cur
treatment and restrict, instead, our spinorbitals to be
centered on a single atomic site. So each "molecular" spin
ortital hill te associated unambigucusly to a specific atom.
We wish now to simulate an a+ominmolecule partitioning
of the pseiidoeiqenvalue problem in (115). To effect the
partitiorinq, let us lock at the diatomic molecule AB at
very large interatomic separation where the fifth and sixth
terms in (TI1) become negligible and the eigenfuncticns of
F in (II2) approach the atcmic cancnical HartreeFock spin
crtitals associated with the twc eigenvalue problems
A A 2 A 1 A A A
F [0l = (1/2 5 2 r + J [C] K [0 ] ) [0] (II6)
i 1A i
E B 2 R 1 E E B
F t rCi = (1/2 V 2 r + J [C] K [0] )} [0] (117)
i 1i i
where the [01 derotes this separatedatcm case (dropping the
reference to clectrcn 1 on the cperatcrs and the functions).
It should be cnphasized tat this limiting feature is a
consequence of the localized form which we have chosen and
that the canonical aclecular crhitals would have separated
rather unpredictably (particularly where high synmetry
exists). Now as we allow the two atcms to approach each
cther and interact, let us focus cur attention on a specific
atom, say A. Suppose that instead of solving (II2) fcr all
of its eicenscluticrs, we seek cnly those scluticns which we
associate with atom A, while the solutions of (117) for
atom B are assumed tc suffice, for the moment, as the
solutions of (112) associated with atom E. that is, we
wish to solve *he pseudceiqenvalue problem
A 2 A A A
F ( [ 1] = 11/2 5 2 r + J [1] K [ 1
i IA
B 1 E B A
Z r + J fC] K [f] ) 4 [1] (118)
1B i
for all the functions centered on A, while holding the
functions on B fixed as though they were already self
consistent eicenfuncticrs cf F in (112). The [11 here
denotes the first iterate of this higherlevel self
consistent process. In order fcr the functions on toth
centers A and B to te sinultanecus eigenfunctions of F, they
must be rcninteractinq through F, that is,
10
Ab E
<4 [ Iis [0 ]> = C, o i,i (IT9)
i i
must be satisfied. It is sufficient, though, that they be
crthcqcral,
A B
= C, v 1, (iII O)
i i
if either is in fact an eiqenfuncticn. But before we treat
this restriction on the eigenvalue problem, let us follow
through kith cur preposition a little further.
We obviously could have just as easily chosen tc
concentrate on atcr P and ended up with the eigenvalue
problem
B 2 B 1 B B
F 11 = 11/2 i 2 r + J [1] K [ 1]
1 1B
A 1 A A B
Z r + 31 [C] K [0] } 4 [1] (I111)
1A j
subject tc the ccnstrairts
A B
< 0 1 4) [ 11> = c, v j,i (ii12)
i 1
This would cive us a full set of functions, t[1]'s,
corresponding tc our first level of iteration. It is fairly
easy to see hcw cne could continue this reasoning to chtain
a set of 4F21's from the ([ 11's, and so on, until scme
degree cf convergence is attained, that is, until
11 
A A
S fnl > I n1],
i i
E E
4 [nl > 4 rn1], (II13)
i 1
and we satisfy tte crtlcqcnality conditions
A B
<4 rn]i r > = C, V ij. (II1)
i i
Getting tack now to the frcblem of handling the
additional constraints on the pseudoeigenvalue protles, we
intend tc treat this problem with the projection operator
technique (Ldwdin, 1961). The proceeding development and
much of wuat is about tc follow could just as well have teen
treated with rartitioninq technique (Lwadin, 1964), tut we
shall disFense with this equivalent approach fcr the time
tinc. let us start by considerirq a general unconstrained
function whose projection is the desired function,
A A (B) B A B
= 4 > < I >
i i i 1 i j
A
= (1P ) 0 (IT15)
B i
where
+ (E) 1 E
P = F = 1 >
E E i i 1
is the prciection operator which projects cut the "BFart"
of the furccticrs on A. Condition (II1Q) in terms of the
12
unconstrained fi actions (dropping row the [n]notaticn at
selfconsistercy) becomes automatically satisfied,
A B A + B
<4 IFI > = < I (1P ) Fir >
i 1 i E 1
A E A + B
< IFI< < P F 14 >
i j i B i
E _A E (B) P _A B B B
= 8 <) $ > 1 S < 1$ ><0 14 >
i i k k i k k j
= 0 (1117)
where we have made usF of the fac+ that the functions on
center B are eicenfunctions of F and thus orthoqcnal arcngst
themselves. The eiqenvalue prcblem associated with atom A
in terms of these unconstrained functions would be
A A A
F(1F ) 4 = 8 (1F ) 4 (1118)
E i i B i
and the secular protlei cwuld then become
_A + A
1<4 I 1P ) IFG)(1P ) It > = 0. (1119)
i B B j
Thus, we have transferred the restriction on the
eiqenfunctions to a modification of the operator. We
therefore seek instead the unconstrained solutions to a
jrojcSfld Haniltonian. Tha analogous secular problem
associated with atcn E wucld be
B + _B
<4 I (1P ) (F )(1P )1 >1 = C (1120)
i A A i
13
with analcqous definitions fcr tho unconstrained functions
on atom E and for *h projection operator associated with
the "Apart" cf AB.
If we rewrite equaiicns (IT19) and (TT20) with the
unprotected operator F separated out, we get (Weeks et a].,
1969)
A _
I1 = C (1121)
i A j
and
B E
I<  IF+V 814 >1 = C (1122)
i B 1
where the pseudcpotentials are qiver by
V = (P FP ) (2r ) [F,P ] (1123)
A D B B P
and
V = P FSP ) (2P ) [F,P ]. (1124)
P A A A A
It should be noted that the second term in (1123) and
(1124) will vanish if the operator F ccmmutes with the
prciecticr operators, that is,
rF,P 1 = rF,F 1 = C. (II25)
A 9
Furthermore, if the "projection" operators are truly
idempotert, then these pseudccotertials reduce to
V = IF6)P (IT26)
A E
and
v = )P .
E A
(II27)
As the Frotlem has beer formulated sc far, both of these
requirements are fulfilled, but these terms shall be
retained for reasons which will he clarified later.
At this point the diatomic problem has been effectively
reduced tc twc coupled atomic problems without any serious
approximations outside the independentparticle model or
limitations iupcsed on the basis, but, we still have not
really simplified the problem with respect to size or effort
either. We have merely reformulated the problcr tc FrcFpt
new insiqtt. The fact that we have not yet gained anything
is evident when we realize 'hat the Coulomb, exchange and
proiecticr operators are all defined in terms of the
constrained functions and that the unconstrained functions
are neither eiqerfurcticns cf F rcr crthoncrmal. Although
the unccrstrained eiqgnfunctions cf our modified operators
A A
(F+V ) 4 = 4 (II28)
A i i i
_E E B
(F+V ) 4 = F 4 (II29)
B 1 1
share a common set of eigenvalues with the constrained
eicenfuncticnn of F (Weeks et al., 1969), they are not
simultaneous eigenfuncticns cf the same operator. Since the
eiqenfuncticnE cn different centers correspond to different
operators, they need nct be orthogonal. In fact, because of
the energy deperdence cf the pseudopotontials, all the
15
eiqenfunctions should, in principle, correspond to different
operators. This energy dependence of th: pseudopotentials
will be taken up in the next chapter.
In principle, one could solve directly for the
unconstrained eicerfurcticns of 'he modified operator and
then reqenerate the constrained eigenfunctions cf F, from
which one could obtain new Cculcmb, exchange and projection
operators (and a total energy). However, this reintrcduces
multicenter additions tc our otherwise singlecenter
unconstrained functions, and thus multicenter integrals,
which is precisely what we have set out to avoid. Ihis
trincs us to cur first najcr approximation.
In the beginning, we assumed that each spinorbital could
be unambiguously assigned to a particular atom and, rcre
importantly, expanded in a singlecenter basis. Cur
unconstrained eiqenfunctions have this property imposed on
them directly by limiting the basis in the solution cf the
projected eiqge value problem. However, as %e have just
noted above, this indirectly forces the constrained
eiqenfuncticns cf the unprojected eigenvalue problem to
deviate from this distinction. Before making any mcre
assumpticrs about the individual spinorbitals, we must
recall that the cnly thinq which has any direct consequence
is the fcrm of the total density itself,
16
(A) A* A
S<4 4, (1130)
i ii
which occurs in the Cculomb and exchange operators, the
proiecticn operators, and the total energy expression. In
terms of the unconstrained functions the total density vay
te writter
(A) A+ A _A
(1P ) i (1P ) (IT31)
i B i B i
which upcn expansion of the projection operators gives
(A)_ A*_A A) (B) _A B B*_A
< o 14 > 4 0
i i i i i i
(A) (B) B _A _A* E
<0 10 > 4 0
i 1 1 i i 1
(A) (B) _A B _A B*
+ >O < I 1 > 4 (1132)
i j,k k i k j k
If we now invoke the Mulliken apprcximation (1949) for the
differential overlap,
*
4 = <# I4 >( + 4 )/2, (133)
i ti 1 1 i i 1
then (1I31) feccmes
17
(A) A*_A (A) (B) _A B 2 _A* A B* B
i> 44
(A) (B) _A B 2 B* B
i 1 i i 1 1i
(A) (B) _A 2 A*_ A
i 1 i i i
(A) A* A
= 4 *'4 (IT34)
i i i
where
A (B) A B 2 1/2 A
S' = (1 <4 (4 > ) 4 (1135)
i i ii i
and
A A
<4 'I '> = 1. (IT36)
i i
Thus, the rulliken approximation leads directly tc a fcra cf
the tctal density ir terms of the sinqlecenter
unconstrained functions which have been renormalized to
unity. So with this as an incentive, we shall replace the
constrained functions, wherever they occur, ty their
renorralized unccnstrained ccunterFarts, that is,
A A
4 = 4 '. (1137)
i i
We will drop the "prime" nctaticn and keep in mind that this
substitution shall always be in effect. The full impact of
retaining the localized singlecenter expansions will not be
realized until apprcximate fcrms fcr the interatomic
interactions are introduced in chapters IV and V.
18
Before looking in rcre detail at the pseudcpotentials,
let us ccrsider what additional problems we can expect when
the same reasoning is extended tc molecules teycnd
diatomics. In brief, the secular problem, analogous to
(1119), that would te associated with atom A in the
triatomic molecule ABE' is
_A + + A
I<( I (1P ) (1P ) (Fb) (1F ) (1P ) I >1 = 0 (TT38)
i B B' B' B 1
where nou, in qrereal, the projection operators are not
crthoqonal (or disjoint), that is,
(E) (E') B 8' B' B
P 4 (1) = <4' It >< (4> 4 (1) (IT39)
E B' j k 1 k k i
where
E E'
1 0. (II40)
1 k
We would qgt analoqcus expressions for atoms B and B'. The
correspcndinq pseudcpotentials woull. thus te mcre
complicated than those encountered in the diatomic case,
unless sceu simplifying assumptions can he made. An attempt
will be made in the next chapter tc retain a pseudopotential
which is accurate to at least secondorder in interatomic
overlap. This next chapter is devoted to a more detailed
lcck at the pseudopotentials, with particular attention paid
to the ererqy dependence.
CHAPTER III
THE ENEEGYCEPFNDENT F5EUDOPOTENTIAL
In the previous chapter, ie have essentially defined an
effective atominmclecule oneelectron Basiltonian. Its
associated eicenvalue prchlem has the fcrm us that
Q FQ 4 = G t (III1)
where F is the polyatcmic analog of (112) and Q is a
product cf prciecticn cperatcrs,
C = (1P ) (1P ) (1F ) ..., ( II2)
B B' B"
which are not in general disjcint (and so Q is nct in
general a protection cperatcr itself). If he expand Q,
C = 1 ... + P + F P + ..., (III3)
B B' D P' B B"
and retain only terms up to secondorder in differential
overlap Iseccndcrder in P), and then invoke the Mulliken
aprrcximation (1S9q) for 'he terms involving interatomic
differential cverlap,
B B' F B' B B B' B'
14 ><4 I = 1/2<4 14 >(, ><4 1+14 ><4 1) (III4)
i i i 1 i i j1
we get tbe qereral form for Q
19
20
C = 1 P (TI5)
where
(E) (B') B B' 2 U B
P = (1 <4 14 > )1j ><4 1. (IIT6)
i ii i 1 i i
The first sur in (ITI6) is over all normalized
(noncrthcqonal) unconstrained pigenfunctions associated with
all the atoms (B) in the molecule except the one with which
C is associated (A), and the second inner sum is over all
atoms except A and the particular atom B being summed over
in the outer sum. The operator P thus "projects" out the
space corFlemenlary to the atcm beirg considered (A), which
is just a generalization of the projection operators we
encountered for the diatomic case in the previous chapter,
that is,
+ (B) B B B
F = P = 14 >
i i i i
with
E (1') B B' 2
y = i <4 14 > (1118)
i jii i 1
being an overlap correction to the "projection" manifold.
In the limit of zerooverlap, F is a true projection
operator with the idempctency relation fulfilled.
The secular problem which we their wish tc solve is of the
saie forn as (1119), namely
21
1< I 11P) (F8) (1F) I >1 = C (III9)
i 1
or in terns of a pseudo Fctential
II = n (III10)
i 1
where, analoqcus to (1123),
V = (FFE( ) (2P) [F,P]. (III11)
In order to simplify the form of V, as in (II26), we need
to examine FP and PP in scme detail. let us first consider
the former,
(B) B B B
FFF = Y l 1 ><0 I. (III12)
i i i i
We must recall ttat the eigenfurcticns in (T1112) are not
eiqenfuncticns cf F but rather
B 0D B
(F+V ) 4 = 6 4 (IIT13)
i i i
with the corresponding pseudopctential associated with the
environment of atom B. Rearranging (III13) gives
S B E B E
F 4 = t V 4 (11114)
i ii i
which through firstorder in perturtation theory becomes
B B E
F = G < (III15)
i i 1
where
E E B E P
S = 8 <4 IV 14 >. (TII16)
i i i i
Using thij now in (II112) qives
22
(13)_E E e
FP = 6y I >
i i i i i
and since F is a selfadjcint oFeratcr, (11115) also grants
FF = FF (III18)
causing the commutator in (III11) to vanish, leaving us with
V = IFg) (2PPP). (11119)
Lockinq ncM at FF we have that
(D ) (E') E E' B E E' 3'
PP = Y y ><4 14 >
i i i 1 i i i j
If we again irvoke Mulliken's aFprcximation (IIT4) then
(B)_E E E E
PF = I y Y i >
i i i i i
where
E E (E') E' B B' 2
y = + t y <4 14 > (III22)
i i jti i i j
Sutstitutinq (11117) and (III21) into (TIT19) we obtain
(E) _E E E F E
V = (E )(2y ) y I ><0 I (ITI23)
i i i i i i
which in the limit of an orthonormal "projecticnn" marifcld
reduces tc the lFillipsKleinmar (1959) pseudopotential
IE) E E B
V = (&_ )14 >< I. (ITI24)
i i i i
Cne should ncte the explicit dopendenc of V on the
enerqy eiqenvalure F. This energy dependence is the crly
remaining ccrFlicaticn in writing down the final matrix
23
equations for solving the secular problem in (IrT10).
Before qettinq irto the energy dependence, though, let us
introduce a basis and proceed tc set up these matrix
eaqations.
For the sake of clarity and simplicity, we consider an
crthonor al tasis centered on the atom (A) with which we are
currently concerned, such that
= u C (11125)
i k=1 k ki
cr in matrix notation
4 = u C (III26)
where M is the number cf functions in the basis (and so the
maximum number of eiqenfunctions we can obtain). We then
consider a compcsite basis for all the other igenfunctions
centered cn their respective atomic centers, that is,
E 1B) E 9
S= uC (III27)
i k k ki
or in matrix notation
E E B
4 = u C (III28)
If we now define the matrices
B
S = , (III29)
kl k 1
(B) _B B E B*
P = (2 )Y C C (III30)
kl i i i ki li
and
(B)_B _B B B B*
R = % (2 )y C C (III31)
kl i i i i ki li
then
V =
kl k 1
t +
= E (SES ) (SES ) (IT32)
kl kl
Thus, asile from the energy dependence of V, the secular
prctlem (III10) reduces tc finding the unitary matrix C
such that
+
C (F+V)C = & (III33)
where
F = (TIT30)
kl k 1
and & is now a diaqcnal matrix.
As already stated, the only remaining complication is the
explicit energy dependence of the pseudopotential (III32).
As long as this energy dependence is there, we can not seek
simultaneous eiqenfunctions of the same operator, since each
eiqenfunction wcula correspond tc a different operator
dependent upcn its cwn eiqenvalue. Such eigenfunctions
would net even be orthogonal tc each o4her. Cperatcrs of
this nature, which correspond to a onedimensicnal filtert
space, are also inherertly rcnHermitear in any matrix
representation (cf order higher than one). We shall now
25
derive an approximate fcrm for the pseudopotential which is
explicitly energy independent and Hfrmitpan. First, let us
rewrite (11133) in terms of a modified Fock matrix, that
15,
+
C FC = 6 (III35)
where
F = F+SPR, (IIT36)
t
P = SES (11I37)
and
t
B = SES (ITI38)
Then we note that from (III35)
PC = CC. (1139)
Noh putting (III36) back into (TIT35) we have
4 + +
C FC+C PCGC BC = (III40)
which upon substitution of (TTI39) becomes
+ 4 +
C FC+C PFCC BC = (III41)
or
C (FEFR)C = (III42)
So ccmparinq this with (11T35) we have that
F = F+1FB (1II43)
which upFc rearranqinq qives
1
F = (1P) (FR) (IJI44)
26
If we now expand the inverse, we generate the series
F = FF+PFPB+PPF... (III45)
frcm which, alcnq with (III36) and (II43), we gather that
SP = iF = EFEP+FPF... (IIT46)
which is similar tc a perturbation expansion. Since the
pseudopotential matrices (II37,II38) are already seccrd
order in cverlap, this series should LP rapidly convergent.
In fact, we shill take crly the first two terms of (III46)
in our approximation. The nonHermiticity is also apparent
in this expansion form, and so we choose the approximate
Hermitean form for (III46)
SP = (EFPF+FPRP)/2. (1II47)
Our modified Fock matrix then becces
F = PJ+(FFBP+PFP) /2. (TTI48)
We have at this point arrived at a scheme for treating a
specific atom within a pclyatomic system. This scheme can
te used fcr each atcm in the system, which are then coupled
in a selfconsistent manner as described in the previous
chapter. All aspects of the HartreeFock scheme (cr any
generalization of this independentparticle model) have been
preserved, with the environment cf each atom being reduced
to the inclusion of additional potential terms. Our largest
sacrifice was trade in return for the onecenter expansion of
27
the eiqenfuncticrs; however, we have not yet used this to
its full advan+aqe. Although we have eliminated all three
and fourcenter twoelectron irtegrals and many twocenter
inteqrals, we have only replaced them by just as many cre
and twocenter inteqrals. This reduction alone may very
well have beer worth the sacrifice, hut the goal here is for
a Tore sctstantial reduction in work. The localized picture
which cones out of this projected Ilamiltonian approach can
be used to its fullest advantage in approximating the
interatomic Cculcmtic and exchange interactions which now
have their closest resemblance tc an electrostatic ncdel.
This will be the topic cf the next two chapters.
CHAPTER IV
A pCDEL APERrCXICATION TC TFF PFECTIVE
CNEELECTRCN INTERATOMIC CCUICMB PCTFNTIAL
If one can substantially reduce the rubber of multicenter
twceloctron inteqrals encountered in conventional analytic
at initic methods, the cciputatiora savings would be
overwhelrinq. In the last chapter, the multicenter
integrals were all reduced to at mcst twocenter integrals
by restricting tie eiqerfuncticns to cnecenter expansions.
The actual number of integrals, however, has not been
changed. We shall in this chapter exploit the localized
picture and sutstartially reduce the number of twocenter
Cculcmb integrals. This shall be done ty replacing the
effective oneelectron cperatcr, for the twoelectron
Coulomb interaction, by a rcdel approximation.
The irteqrals which we wish tc approximate are of the
form
E
<4 l1J K > (IV1)
i i
28
which is the Coulombic interaction of the ith spinorbital
with all the ctter spircrhitals in the polyatomic system
except thcse on the same atomic center (A). The effective
oneelectron operatcr is given more explicitly by
E (U) F E 1 P
J = q <4 r d1 > (IV2)
1 i 1 i 12 i
where the q's are spincrtital occupation numbers and the
subscript 1 is used (but dropped from hero on) to emphasize
the functional dependence of the operator J en the
coordinates cf electric 1. When the eigenfunctions are
expanded in a basis, one generates the multitude of
multicenter twoelectron integrals cf the type
E 1 U
. (IV3)
k m 12 1 n
It is the intricate dependence of J on the coordinates of
electron 1 that does not allow fcr a simpler treatment of
this operator, and so we seek an analytic function with a
simpler functional dependence tc approximate the operator J.
Let us restrict ourselves tc the case of a diatomic
molecule for the moment, where J is centered on a single
atcm (B). Then fcr large interatomic separation J
approaches the approximate form
E E 1
J = b r (IV)
30
where the total electronic charge is reduced to a point
charge en the distant atomic center. This corresponds tc
replacinc the electronic coordinates of electron 2 in (IV2)
by the nuclear coordinates of atom P, that is,
E B) E E 1 F
J = q <4 It >. (IV5)
i i i 18 i
At lesser interatomic separation, where the charge density
can no longer te considered a point charge, the
interelectronic distance must undergo an effective dilation
to account for the mere diffuse charge. Furthermore, as
electron 1 aFprcaches the nucleus of atom B, the nuclear
attracticr should net witness any appreciable screening, and
therefore any model function should go to zero as this
distance goes to zerc. The following model potential
function, which we now choose, has these desired properties:
_E E D 1
J = F r 1Y (r ) Ir (IV6)
1B 1P
where X is a screening function with the asymptotic behavior
X (0) = 1 (IV7)
and
E
X (oo) = 0. (IV8)
Such a screening function is inherent to the ThomasFermi
model (Thomas, 1928; Fermi, 1928) of the atom. A
sionificart feature cf the ThorasFermi screening function
is that it is universal for all neutral atoms with respect
to the dinensicnless variable
31
2 E 1/3
x = r (3n/8) Z ] (IV9)
1E 1B
in terms of the nuclear charge. The distance r is assumed
to be given in Bohrs. Pctentials of the form (IV6) are
certainly nothing rw and most such potentials (Hellman,
1935) adopt ar exponential behavior for the screening
function X. he rake the choice
E > F P E E PB 
X (r ) = A X In 1 m a ;r ) (IV10)
1B t t t t t t t ID
where
p  n1
X (n,l,m,a:r ) = r exp(ax )Y (6 ,a ) (IV11)
t 1F 1F 1E Im 18 1B
and where te expenertial factors are, in principle,
universal and could he used for all atoms (with the
appropriate change in x). Any angular distortions are taken
into accccnt with the real normalized spherical harmonics.
The linear coefficient are to be determined by fitting the
approximate form of J in (IV6) tc its true form in (IV5)
alona with the constraint in (TV7). We have merely
expanded the operator J in a basis of Slatertype functions
(with a somewhat universal choice cf exponents).
The lirear coefficients are computationally guite simple
to handle since they can be taken outside the integrals in
which they appear. The coefficients can te determined by
substituting (IV6) for (IV5) in the onecenter Coulomb
inteqrals cccurrina or atom B and matching their values,
that is,
B _ P B B P
<4 IJ 14 > = < IJ II t > (IV12)
i i i i
for each eiqenfurction on atom B. In more detail we have
F E > B D B  1 B
N r<4 Ir It > A <4 IX (r )r 14 >]
i 1B i t t i t 1P 1B i
(P) B B. B 1 B P
= q <4 t Ir 14 4 > (IV13)
1 1 i i 12 i i
%here the Coulcrb integrals on the righthand side cf the
equation are cnecenter inteqrals which we properly compute
and retain. he have as many equaticns like (IV13) as there
are eiqerfuncticns en that atom, consequently we can fit up
to just as many linear coefficients. The number of terms
that are needed in (TV1C), though, are usually much less
than that (due tc its relatively smooth behavior), sc scme
type of weightedd) leastsquares fit procedure should he
adequate. One other condition, perhaps even more important,
that can be satisfied is the +ctal twccenter nuclear
attracticr experienced by that atcm, qivon by
A (B) B F 1 B A >
Z q <4 Ir I > = Z J (r ). (IV14)
1 i j 1A i BA
In the general case of a polyatcmic, this would be a sum
over all the other atcms. Thus, in addition to the
equations in (IV13), we can have
32
33
B 1 B > 1
N fr A X (r )r 1
AE t t t AB AB
(B) B E 1 D
= q < E Itr 1t >. (IV15)
1 i 1A B
In order to determira the linear ccefficients, these linear
equations can 1'e written in the matrix form
T A = G (IV16)
where
E
A = A (IV17)
t t
B E B 9 1 B
T = 1 <0 IX (r )r !( >, (IV18)
it i t IP 1B i
E B > 1
T = N X r )r (IV19)
M+1,t t AE AB
T = 1 if 1 = C,
M+2,t t
E
= 0 if 1 > C, (IV20)
t
E B 1 E
G = N
i i 12 i
(B) E E 1 BE
q < 4 Ir 1i 4 >, (IV21)
i i i i 12 i 1
B 1 (B) B B 1 B
G = N r q , (1V22)
M+1 AB 1 1 j 1A i
and
G = 1.
(IV23)
311 
In general M+2 is larger than the number of terms needed in
(IV10), and so we have an overdetermined set. If M+2 and N
were equal, then T would he square and
1
A = T G, (IV24)
provided that T is nonsingular (no linear dependencies).
However, in the general case, one can write the normal set
of equations IIV16) as
t
(T vT)A = IT w)G (IV25)
where w is an arbitrary diagonal matrix whose elements act
as weiqhts +c the original M+2 equations (IV16). The
choice of a urit matrix cwuld ccrrespcnd to the usual linear
leastsquares fit procedure; nevertheless, the option of
giving more importance to certain conditions such as the
external ruclear attraction (IV11,15) or to the valence
orbitals ever the core orbitals can be useful. Since we now
have square matrices, we can solve for A in terms of the
generalized irverse of 1, that is,
t 1
A = (I wT) (Tw) G. (IV26)
The exponents in (TVlO) could be determined once and for
all by a mere fit to the universal ThomasFerri screening
functior nenticned earlier, hut this would build in the
deficiencies of the ThomasPermi model (particularly for
small atcas). A better approach, perhaps, was suggested by
score wcr< carried cut by Csavieszky (1968, 1973) He
35
performed a variational calculation to determine an
analytical solution to the ThorasFermi equations with the
following trial function for the screening function:
'x IX a"x 2
X (x) = (A'e A"e ) (IV27)
His calculaicn yielded the exponents
a' = C.178255C and
a" = 1.759339 (IV28)
as well as values for A' and A" (tut these are nct cf any
concern to us). This ccrresponds to the following choice of
exponen+s in cur model potential expression (TV10):
a = 2a' = 0.3565118,
1
a = 2' + a" = 1.937594, and
2
a = 2a" = 3.518678. (IV29)
3
These values perc derived for spherically symmetric neutral
atoms in the ThomasFermi model with somewhat modified
boundary conditions iallied by the particular choice of
(IV27). In crier to allow for corrections to the Thoras
Fermi model, radial and anqular distortions, and deviations
from neutrality that would exist for an "atomina
molecule", some additional flexibility (other than the
variation of the linear coefficients) might be necessary.
Auqmentinq expression (IV10) with an extra term or two
would probably l sufficient for most cases, but angular
distortions would certainly necessitate the inclusion of
some polarization terms. One could also just choose some
36
qereral iellrourdel, flexible basis, such as an even
te~perpd (Bardo and ?uedenberq, 1973) set of exponents, with
the hopes that thp use of the dimensionless variable x and
welldetermined lirear ccefficierts aculd he sufficient.
CHAPTER V
AN AFFECXIM'AT NCNLCCAL INTEPATOMIC
EXCIHANGF PCTENTIAL
In the last chapter, we substantially reduced the number
of thccenter twoelectron Coulomb integrals by exploiting
the localized picture. We shall now attempt a similar
reduction in the analogous exchange integrals. Again, we
wish to approximate the effective oneelectron operator, hut
this time the operator is ncnlccal and requires a somewhat
different approach.
The tccerter excharqe integrals are of the form
<4 K 10 > (V1)
i 1 i
where K is the effective nonlccal oneelectron operator
subscriptt 1 dropped from here on) for the exchange
interaction with all the other spinorbitals (of same spin)
in the pclyatcmic system except these associated with this
center (A). Ihe operator K is defined such that
E (E) E 2 1 B
K q4 (1) = q <4 tr I > ) (1). (V2)
i j1 j 12 i 1
38
Let us consider the asymptotic behaviour of (V1) when the
average irterelpctrcnic distance approaches the internuclear
distance, that is, %hen
E 1 (1) B B E
< 1K 14 > = r q <4 104 > (V3)
i i AB 1 1 j i i j
or
E 1 (B) B B B
K = r q I( >
AB i i I i
Since K is nonlocal, we can no longer use any of its "local"
properties, such as its expectation values with respect to
the eiqerfuncticns on its own center, to help determine an
intermediate forn tcr its approximation. Cur endeavor thus
far has teen to retain a theory which is valid through at
least secendcrder in diatcmic cverlap, and so it is felt
that retaining K in its asymptotic form (V4), which is
secondorder in overlap, would not be inconsistent with any
cther apFpcxirations made thus far. In such a case, the
interatomic exchange can be treated with the same ease as
the pseudcpotential in chapter III.
This approximaticn may appear to be a bit crude at first,
but the naqnitudp of these interacticns in this localized
scheme is relatively small due tc the nearly electrostatic
nature of the model. It should be emphasized that the
approximations used for the interatomic twoelectron
interactions could only be realized in such a localized
separable picture.
CHAPTER VI
4 MOECL HAMIITONIAN AND AN APPROXIMATE
ELECTRCN ERCFPGPTOE
According to the Hiisenberq equation of motion fcr the
electron propaqaeor (Linderberg and Chrn, 1973), in the
energy representation, we have ir matrix form that
G (E) = <
>
ii i i "
1 + +
= r[<[a ,a > + <<[a ];a >> ] (VI1)
i 1 + i j F
where
a = <4 I~ > and
i i cp
a = <4 14 > (VI2)
i op i
are the annihilation and creation operators, respectively,
uith the anticommutatio relations
+ +
[a ,a ] = [a ,a 1 = 0 and
i 1 + i 1 +
4 4
[a ,a 1 = a a a 1 = = <4 It >, (VI3)
i 1 + i + ii i j
39
40
defined in terms of field operators and a spincrbital
tasis. The ianyelectrcn Hariltcnian, in second quantized
fcrr, is
+ + +
H = h a a 1/2 2 (klij)a a a a (VI4)
ii ii i i ijkl i k 1 j
where
h = <  h it > (VI5)
ii i 1 1
and
1
(kl(ij) = <$ 4 Ir I4 > (VI6)
k i 12 1 j
are the usual cre and twoelectrcn integrals. Equation
(VI1) can be iterated to yield
1 2
G (E) = S t F F + . (VI7)
ii ii ii
where
+
F = (VI8)
ii i i +
is the first acrent and sc on. Substituting fhe Hamiltonian
(VI4) into (VI8), the first moment becomes
+
F = h + r (iilkl)(illk ) ] (VT9)
ii ii kI k 1
which has the same form as the effective oneelectron
Hariltonian as originally presented by Fock (1932) and will
hencefortl be called the Fcck matrix. This more general
apcrcach to HartreeFcck theory will be taken up in the next
chapter.
41 
Suppose now that irstead of using the correct many
electron Familtcnian, one substitutes into equation (VI8)
an approximate oneelectron model Hamiltonian of the fcrm
+
l = Eaa a = n (VI10)
k k k k k k k
where the n's are just the occupation number operators and
the S's are real, negative energies of the noninteracting
electrons. Tr cur present treatment, we would choose as our
spincrbital basis the nonorthoqonal atomic spinortitals
generated from each projected Hamiltcrian calculation, and
for the energies we would choose the ccrrespcnding
eiqenvalues, with possible modification (to the positive
eiqenvalues, fcr example). The Fock matrix then takes the
approximate fern
F =
ii i i +
= C S S (VI11)
k ik k ki
or the matrix fcrm
F = SES (VI12)
where 6 is a diagonal matrix. Equation (VI7) in ratrix
form teccaes
1 2
G(E) = E S S S E .
1 1
= (ES 6) (VI13)
where one car see that the pcles cf the propagator will
cccnr at the eigenvalues of the Fock matrix, that is, at the
zeros of the secular determinant
ISSSF I. (VI14)
The ccrrespondinc eigenvalue Frcblei
PC = SCE (VI15)
can te reduced to the simpler one
F'C' = C'E (VT16)
where
+
C' C' = C'C' = 1 (VI17)
and
F' = ]SX (VI18)
with X being the diagonal matrix with elements
1/2
X = (8 ) (V119)
kk k
This can be considered as the diagonalization of a Fcck
matrix in a basis which is energyweighted ldwdin
crthoqonalized, and this is the reason for the term "Energy
Weighted maximum Cverlap" (EWMO) used ty Linderherg et al.
(1976) to describe this methcd. The molecular orbital
coefficierts in the original basis are given by
1/2
C C' (E /F ) (VI20)
ki ki i k
The electron propaqator in this orthgconal tasis becomes
1 1
G(E) = X (EI1+XSX) IS, (VI21)
and the elements cf the corresponding oneelectron reduced
density ratrix are given by the contour integral (LinderLerg
et al., 1976)
+ 1 +
= (2i) ) <>
k 1 C k 1 F
1
(2ni) dEf G (E)
C k1
1i 1 1
= (2ni) d (X (21+XSX ) XS) (VI22)
C kl
where C enclcses the occupied energies. An appropriate
defiriticr of t1e carqe and bordcrder matrix iculd in this
case be
1I 1
P = 12ni) ) IF IE1+YSX) (VI23)
C
This choice leads to
2
q = F =
k kk i i ki
for the formal spinorbital charges in terms of the
molecular orbital occupations. This definition is
equivalent to the orbital populations according to Mulliken
(1S55).
In the case cf positive energies in (VI10), one would
have tc resort to some other type of orthogonal
transformation in order to avoid a complex Fock matrix
(VT18). Lewdin's (1970) symmetric orthogonalization
procedure would mcst closely resenhle the procedure just
described. The resulting population analysis is, of course,
dependent on the type of crthcgcnal transformation used
unless the analysis is done with respect to the original
nonorthoccnal Lasis. Pcpulaticn analysis ir a nonorthogonal
basis, however, can lead to unreasonable charges (negative
or greater than cne). One could, of course, always solve
the complex eiqenvalue problem and still uise the ateve
procedure, but, in general, the irclusicn of positiveenergy
(virtual) spirorbitals is not a straightforward task.
CHAPTER VII
THE GCPNPCANCNTCAL HARTBEEFOCK SCHEME
The notion of an atom in a molecule led Slater (1970) to
consider an enqrny functional obtained as an average over
multiplts arising from a given configuration. His
extension of this idea to configurations with fractional
cccupaticns, known as the HyFerHartreeFock mathcd,
however, has been subject to some criticism because cf the
appearance of cffdiaqcral Lagrangian multipliers and lack
of certain conceptual grounds (Jargensen and Mhrn, 1973).
In this same spirit we wish to introduce an average based on
the statistical mechanical concept of an ensemble (Abdulnur
et al., 1972). The particular choice of a Grand Canonical
ensemble, as defined by its density operator, leads to the
eliminaticn of offdiaqcral Lagranqiar multipliers.
A system of noninteractinq electrons described by the
Hamiltcnian
H = n (VII1)
i i i
45
46
can Le described by the Grand Canonical partition furctior
related tc the density cFerator
S= n (1n +2 n )/(1+z ) (VII2)
i i ii i
%here
z = expf (& p)/T]. (VII3)
i i
The parame+e rs p and T are the thermodynamic chemical
potential and absolute temperature. The average value of an
operator A for such a system is then given in terms cf its
trace witt respect to the density operator, that is,
= 'r(Af ). (VTI4)
In particular, the average value of the number operator is
= / 1+z ). (VII5)
i i i
Usinq this ir expression (VII2) gives a unique way of
defining the density operator from a given set of occupation
nurters,
p = n rl+(21)n ], (VII6)
i i i i
without any reference to any thermodynamic parameters.
Although there is nc connection with equilibrium situations
in statistical mechanics, we shall refer to this as a Grand
Canonical density operator, which acts as a formal device
for the formation of averages only. Except for the case of
all integral cccupaticn numbers, this form of the density
operator qives a nonzero width (uncertainty) for the tctal
number operator.
17
If one considers an electronccnserving (Cancrical)
average valuE of the nanyelectron Ilamiltonian in the
crthonormal basis which diaqonalizes the density matrix, one
has
= h + 1/2 (J K ) (VII7)
i i i i i1 ij i j
where the only thing which is not explicitly determined is
the average cf the number operator product in the second
term. If, however, th1e average is performed with respect to
the GrandCancnical density operator then
, (VII8)
ii i i
and equation (VII3) can he written
= 1/2 3 V (VII9)
i i i ij ij i j
where the effective oneelectron energy is given by
S = h V (VTI10)
i i i ii i
and the effective interaction energy by
V = J K (VII11)
ii ii i1
Cre recognizes that these energies can te interFreted as the
first ane second partial deriva+ives of the total energy
functional hith respect to the occupation numbers as
follows:
a/a = E (VII12)
i i
48
2
a /aa = a& /a = v. (VIi13)
i i i i ij
Such interpretations lead to the finitedifference
approximations tc icnization energies
(i)
d E = A (VII1)
i i
and excitation energies
(i i)
A E = S + F A + V % L (VTI15)
i i 1 i i i
for which Kocpmans' (1935) thecrem is a special case.
Application of the mean value theorem suggests that the test
approximations to these quantities would be obtained by
determining the P's and V's at some intermediate point, such
as =1/2. Such a scheme has been termed the transition
state method (Slater and Wood, 1971) when applied to similar
energy furcticnals.
qhe effective oneelectron (GrandCanonical) operator
having ttese cneelectrcn ererqies as its eigenvalues, has
matrix elements, in an arbitrary nonorthoqonal tasis, cf the
form
+
S = h + r (kllk'l')(kl' Ik'1) ] (VTI16)
kl kl k'l' k' 1'
where the density matrix elements are given by
+ +
C (VII17)
k 1 i li i ik
and C is the uritary transfcriaticn ratrix in this basis
which diaoonalizes F, in cther words, its columns are the
associated eiqenvectors of F. 1he iterative selfconsistent
construction and solution of equation (VII16) is what has
come to be called the GrandCanonical HartreeFock (GCIF)
method, .ith the total energy given by equaticns (VII9),
(VII10) and (VII11).
CHAPTFR VITI
A SELFCCNSISTENT CHARGE ANE
CCNFIGURATICN PRCCEDURf
Up to now we have said very little about the manner in
which the spincrbital occupations are assigned. This is
somewhat of a sensitive subject, here and in many other
"bcildinqblock" aVproaches. This question has its roots in
the early days of valencebond methods. I e would like to
make an analcqy between our method and the single
confiauration cf nonorthoqonal crhitals methcd.' This
method amounts *c a condensation cf many configurations to
one, built uV from distcrted (hybridized) atomic orbitals,
which, in essence, is wha* Coulscn and Fischer (1949) did
for the tydroqen mclecule. The application of this method
to larger systems, however, is quite limited because of the
difficulty in calculating the ratrix elements ct the
Hamiltonian, sicce there is no crthcgonality between the
'The literature on these methods is just too extensive to
review here, hut we sucqes* Gerratt (197U) for an excellent
account cf the subject as it applies hero.
50
51 
orbitals. The model itself provides a sensible
interpretation of chemical bondicq in terms of a distortion
of the rarticipatinq atomic ortitals combined with a
recouplinc of the spins. The method has been shown to be
capable cf yielding useful estimates of molecular binding
energies and ctFlr properties. He view our method as a
direct reans of obtaining such distorted ncnorhcgcral
ortitals, were the orthcqcnality problem has been
incorporated into the pseudopotentials. The renormalized
pseudoortitals (unccistrained furcr;cns of chapter IT) then
act as these tybrid orbitals. The tctal energy we compute
from these orlitals is valid through secondorder in overlap
(from the Mulliken approximation) and so we should, within
the intqeral apcroximations made, be able to mimic such a
scheme, es far as %e can tell. Thpre are some other basic
differences, however. These methods generally separate the
space and spir parts of their a vefunc tions, and
consequently couple their spins with proper generalized spin
functions.
The concept of "atcric valency" arises when molecules are
allowed tc separate into their constituent atoms. When the
most general linear ccmbiraticn of spin couplings is formed,
the resulting separated atomic states can be regarded as
welldefined "valence states". Any less couplirg, however,
can lead to illdefined rcnccupled states, particularly when
52
degenerate states are involved. This we feel is mostly due
to the separation of space and sp n, connected with an
imbalance of symmetry constraints cr the two parts. In our
treatment we use spinorbitals, which frcm the beginning
puts space and spin cn similar footing. A single
configuration of spinorbitals will, in general, not
correspond to a proper spincoupled state, but the atomic
"valence states" remain rather welldefined, since each
valence electron is accomodated in a distinct (aFart from
exchange) atonic orbital with a distinct spin.
One desires a configuration which allows one to describe
the dissociation process as a smooth recoupling of the
ortitals from a perfectlypaired state to the proper atomic
states. Particularly attractive, in this regard, is the
spinvalence theory eurloyed by Heitler (1934), where
eiqenfunctions of the atomic Hamiltonians are ccupled
together to fcrm the molecular state. In fact, this was the
motivaticC behind Mcffitt's (1951) atominmolecule
approach. The basic question is "hcw do we choose cur spin
orbital occupations in our singleconfiguration, which has
these desired properties, without falling into the traps
that other methods do when all the proper couplings are not
taken into account?" Defore we answer this, let us mention
two methods which come close in spirit to what we are about
to describe. 7he first is Hund's rule coupling (Gerratt,
53
1971), and tie second is alternate molecular orbital (AFC)
theory (Faunc?, 1967). We mention them because of the
physical picture they present; however, in those two related
methods, space and spin are separated, and the resultant
spin functions are complicated. We feel tha+ if one instead
uses spinortitals and relaxes seme of the constraints'
normally imposed on the spin (Gunnarsson et al., 1977), then
the same physical picture suggests a very simple scheme for
making our assiqrments.
The procedure is to start out with a configuration which
corresponds tc a covalent structure that one generates from
valence sell electrcnpair repulsion theory (VSEPRT) as
described in any general chemistry text. The spins are
assigned such that the electrons on each atcm are coupled tc
give maximum resultant and such that the resultant spin on
each atom alternates in sign with respect to each of its
bonded neighbors. This should be done such that the total
resultant spin is the desired cne (if possible). Consider
the example of a carbon monoxide rclecule. It has the valid
structure
tCS" C (VIII1)
corresponding tc the corfiguraticn
'These constraints were necessary in the other treatments
to get proper separation of the atomic states. This is rot
necessary in cur simple treatment.
54
(C s) (C1s') (Cs) ) (C1s') (02s) (C2s') (C2s) (C2s')
(C2px) (2px') (n2p ) (C2pz') (02py) (C2py') (VIII2)
where the prices denote opposite spin. There are of course
ionic structures as well, but we shall incorporate them
later. The main point here is that this structure separates
into the atomic states represented by the configurations
(C1s) (Cls') (C2s) (02s') (02px) (02 x' ) (02py) (02pz) (VIII3)
and
(C1s) (C1s') (C2s) (C2s') (C2pz') (C2py') (VIIIU)
each satisfyirq Hund's rule. This type of structure can be
generated for any system where the atcms can be divided into
tc subsets in such a way that no two atoms which belong tc
the same subset are neighbors tc each other. Such a system
is called an "alternate" system. For nonalternate systems
the situation can be quite different. In fact, in
nonalternate solids this can give rise tc antiferromagnetic
conducticn sheets.' In cyclic molecules (the only molecules
which car be nonalternate) ncnalternacy is often accompanied
by unusually stable ions, radicals, "sandwich" complexes, or
some higher level cf alternacy.2 Fven though this would he
an interesting +cpic on its own, we shall not dwell on it
here. The main point we wished to make here is that withir
each atoi we tave parallel alignaert cf spins (Hund's rule,
IFcr example, an FCC lattice might have alternating planes
cf parallel spin, thus a higher order of alternacy.
ZSuch as sicma and pi planes.
ferromaqretic coupling) and between atoms we have (at least
for alternate systems) antiparallel alignment of spins
(antiferrcmaqnetic' coupling). This is, of course, just a
general cuidelire, and one can make any occupation
assignments ore wants, provided that care is taken not to
break syneotry with exchange polarization (the scheme just
described will not streak symmetry when applied to alternate
systems).
Sc far we have just devised a way to assign spins such
that we cot siccth uncoupling upon separation. Suppose,
though, that this single neutral structure does not properly
describe the molecular state or the separatedatom states.
What if we krew that the molecule had a triplet2 ground
state which can not he formed by a smooth coupling of the
separatedaton qrourd triplet2 states? What if we suspect
the ground state to be a great deal ionic in character?
These types of questions involve what is perhaps best
described as the crossing of different singleconfiguration
diatatic states (O'lalley, 1971). The simplest prodedure
would be to compute hoth of the diaLatic states involved and
"uncross" them. In fact, for the first case where the curve
INot to exclude the possibility cf complete ferromagnetic
ccupling.
2Ey "triplet" we mean two upaired parallel spins; total
spin is never properly taken in*o accountt in our treatment.
3"smooth" meaning no sudden charge transfer or spin flips.
56
crossing corresponds to a spin flip, this is atcut the crly
alternative ore Fas in the present treatment; however, in
the second case we only have charge transfer (no spin fliFs)
taking place. This brings us to the main theme of this
chapter a selfconsistent charge and configuraticn
procedure.
As mentioned earlier there are other valid (ionic)
structures one can write down for carton monoxide besides
the covalent one in (VITI1). These other structures are
nothing more than charge transfers from the covalent
structure (disiissirq spir flips). Ry employing fractional
occupations it cur configuration we can go smoothly from one
structure tc another (one diahatic state to another). In
general, the molecular state itself right test te described
hy such a configuration with fractional cccupaticns
(configuration mixing) which goes "adiabatically" into the
separated atomic states with integral occupations. How does
one determine these fractional occupations? Sc far we have
not even said what determines the molecular state in our
treatment. Cne could just define it in +erms of a given
(valencetond) structure ccrstructed from the atomic
crbitals, but most molecules protatly can not he well
represented by a single structure. In chapter VI we
developed a means cf generating mclecular (delocalized)
ortitals from a model Hamiltonian built up from the atomic
57
ortitals and energies. Perhaps the best way to describe the
molecular state then is in terms of these molecular
orhitals. We could cheese integral occupations for these
molecular orbitals corresponding tc scme desired state and
then let the population analysis determine the fractional
cccupaticns fcr *he atomic orbitals which qc into our single
valencecund structure. Such a prccedure would be very much
like the hypor[artreeFock method proposed by Slater et al.
(196S) fcr crystals. This type of procedure would allow one
to compute a Fctential energy surface which by construction
corresponds tc a molecular state with a specific spin' and
anqular momentumt, as well as to a specified set of
separatedatom states with the sane net spin and angular
cerontum. Perhaps the best way tc illustrate this is by an
example, and the next chapter should serve this purpose.
ICnly the compcnent alcng some corren preferred axis is
explicitly and uniquely specified.
CHAPPEP IX
APPLICICATINS
A. Nitric Cxide Molecule
Despite the important role of nitric oxide and its
positive ion in the upper atmcsphere and its ecologically
undesirable presence in the exhaust emissions cf cur ever
sopopular automobiles dcwn here cr earth, there has teen
relatively little thecretical electronic structure %ork
reported for this firstrow diatomic molecule. Restricted
HartreeFcck and configuraticn interaction calculations have
been carried cut for the qrcund (X2n) and first excited
(A2z*) states of nitric oxide at their equilibrium
qecmetries by Green (1972, 1971) to yield cneelectron
properties with rather limited success.' Put tc our
knc!ledqe their0 has been no unresricted HartreeFcck
calculation (spinprojected or otherwise) reported on this
ISee also the natural orbital calculation of Kouba and
Chrn (1971).
58
59
oddelectron paramagneticc), openshell system tc assess, on
a oneelectror tasis, its complex phctoelectron spectrum
(Turner et al., 197C). Configuration interaction
calculations for the positive icn have been done, however,
in an attempt to assign the various observed states (11
between 9 and 24 eV) by LefehvreBricn and Koser (1966) and
later by 'hulstcup and Chrn (1972) The assignments appear
to be resolved, although alternate assignments have been
proposed ty Ccllin and Natalis (1968).
We make nc attempt here to confirm or challenge these
assiqnmerts, tut rather, to test our theory for its
strengths and weaknesses on a reasonably small but ccplex
system before applying it to a much more ambitious problem.
Since our air is to look into the chemisorption of nitric
oxide on a nickel surface, perhaps we should know how the
theory works on nitric cxide itself. From the outset we
expect (cr at least hope) 'hat the dissociation process, in
terms of the interaction of localized atomic crbitals and
the associated total energy, should be reasonably well
described, since the tctal energy is not dependent upon how
the total density is brcken up into various contributions
but rather on the total density alcne. On the ether hard,
properties associated with the delocalized molecular
crtitals derived frcr our mcdel Hamiltonian, such as
Kcoprans' energies, musF be taken with less reliance. In
connection with this, charge transfer (between valencebond
structures, if you wish) is expected to be a sensitive
issue. hbis too, in our procedure, depends cn the model
Pariltonian used tc generate the rclecular ortitals from
which the population analysis proceeds. For the most part,
we will avoid such complicaticns by restricting ourselves to
single valencebcndtype structures, that is, no charge
consistency will in general be attempted.
For nitric oxide in its ground (X'n) state, we will use
the conficuraticn
(C1s) (C1s') (Nis) (Ills') (C2s) (C2s') (N2s) (N2s')
(C2pz) (N2pZ') (n2px) (N2px') (02py) (02py') (N2py') (IX1)
in terms cf "perfectpaired" atomic spinorbitals where the
primes denote the najcrity spin. This is unitarily
equivalent to a molecular configuration
(16) (I ) (;2 ) 120') (3() (3(' ) (45) (46')
(rc) (P5 ') (1 Tx) (ITnx') (Iny) (1 y') (2ny') (IX2)
where the 26, 46 and 2ny orbitals would be of antibonding
character and the 1i ard 1y ortitals of rather nonbcnding
character. In our procedure, the nitrogen and oxygen
orbitals in (IX1) are determined separately in an
alternating fashion coupled tc each ether in a self
ccnsistent garner as described in earlier chapters. From
these atcric ortitals, a model Hamiltonian is constructed
whose natural crbitals canonicall HartreeFock solutions)
ccrrespcnd roughly to the molecular orbitals in (IX2).
61
The basis sets used for nitrcqer and oxygen are STO's of
doublezeta quality (Poetti and Clementi, 1974) plus dtype
polarization functions, for a total of 15 functions cr each
atcm. The model potentials were expanded in a set of 4 s
type functions and one ptype function polarized along the
bond axis (zdirection here). We chose an eventempered set
of exponents (0.5,1.C,2.C,4.C) scaled by the cube root of
the nuclear charge, as suqqested in chapter IV. The
exponents for the lasis sets and the model potentials are
listed in Tables 1 and 2. No attempt was made to optimize
these exponerts, and the polarization functions were
selected rather arbitrarily.
The first calculation was carried out at the equilibrium
qecmetry cf 2.17 a.u. with the procedure just described
above and in previous chapters, but the spinorbital
occupation numbers were kept fixed. In other words, any
charge transfer predicted from the population analysis
between each cycle was ignored. As stated above, this
corresponds to a single valencebondtype structure. This
is done tc eliminate any effects due directly or indirectly
to the mcdl Hariltcnian. The result was a total energy of
129.44 a.u., which is about 0. 1 a.u. below the estimated
HartreeFcck limit (Hcllister and Sinanoglu, 1966). The
subsequent model (FWMC) [amiltonian calculation, however,
qave molecular crtital energies which were essentially in
Table 1. Easis sets for nitrogen and oxygen.

a a
Type nitrogen oxygen
1s 8.49597 9.46635
Is 5 .9 fU 6.63768
2s 2.26086 2.68801
2s 1.42457 1.67543
2F 3.24933 3.69445
2p 1.49924 1.65664
b b
3d 2.00000 2.00000

a
From the tables of coetti and Clementi (1974).
b
Unoptimized Folarizaticn function.
Tatle 2. Model potential exFonents for
nitrcqen and cxyqen.
a a
Type nitrcqen oxygen
1s 0.9c47 1.COCCC
1s 1.91293 2.00000
1s 3.82 86 '4.COCCO
Is 7.65173 8.00000
t b
2pz 1.91293 2.00000
a
Eventerpered set cf furcticns (C.5,1.C,2.0,4.0)
scaled ty Z**(1/3).
t
Polarization function along tending axis (1.0).
63
reverse cf the ctserved crder for the 56 and 2ny orbitals,
and cave an ancmalcus 3' orbital 5C eV below the observed
value.
In an effort to see what the source of these
discrepancies was assuringq that KcoFmans' approximation is
not all tc blame), a second calculation was performed where
charge transfer was allowed tc take place, that is, charge
consistercy was attained. The result was a somewhat higher
total energy, with some improvement in the molecular orbital
energies, but the tasic discrepancies persisted. The net
charge transfer %is only C.015 (frcm oxygen to nitrogen!),
which suggests that the single (neutral) valencetcnd
structure is perhaps net so inadequate. Could it be that
the model Hatiltonian constructed from just the occupied'
atcmic crtitals is toc pccr?
The next logical step appeared to be an improvement of
the rcdel Hlamiltcnian. As ePnticned in chapter VII, the
inclusion of virtually orbitals in the EWMO Hamiltonian
required either complex arithmetic (a drastic and
undesirable step) or an alternate methcd of treating the
ncnorthoqcnalit' which subsequently alters the population
'With respect +o the todel Haniltcnian, "occupied" merely
implies a negative orbital energy and "virtual" a positive
energy.
60
analysis. The second alternative requires an extra
diaqcnalizaticn of the metric matrix (if L8wdin's (1970)
symmetric orthoqcnalization is used) but is much easier to
implement computationally. This leads to a population
analysis which is, in part, arbitrary if done in the
orthogonal basis (which is necessary *o guarantee reasonable
occupation numbers in a chargeconsistent calculation).
Such a calculation was Ferfcrred. There was a substantial
improvement lut the relative ordering of the orbital
energies has still in disagreement with the observed values
as they feve teer assigned. Charge consistency gave little
mcre improvement. The main discrepancies are ir the
relative order of the various sinqletst and triplets and
the apparent reversal cf the 54 ard 2ny levels.
Cne final improvement, still involving the virtual
ortitals, was attempted. In the mcdel Hamiltonian, as
constructed above, the virtual crtitals play a rcle
equivalent tc that of the occupied, and the energy
associated with each virtual crbital should then be the
energy it would have if it were indeed occupied. This is
related to the problem encountered in configuration
interaction and manybodypPrturbation calculations, where
IThe terms "singlet" and "triplet" refer to 'he final
icnic states resulting in the dele+ion of a majority or
minorityspin ortital, respectively (the "singlet" actually
being an average of singlet and +riplet states).
65
the HartreeFcck virtual crbitals used in +heir expansions
arise from a potential generated by +oo many electrons.
This errcreous potential comes about from a miscancellation
of Cculort and exchange interactions an electron has with
itself. This cancellation is an important artifact,
however, of the HartreeFock method which allows the use of
a single operator fcr all the occupied (and virtual)
orbitals (luzinaaa, et al, 1973). An approximate first
order correction to this can be achieved by just rescaling
the Coulcmb and exchange contributions to the orbital
energies to account for the prcer number of particles
involved in the potential. cor a chargeconsistent
calculation this involves all partially occupied orhitals,
each with its cwn scaling factor. Such a correction was
made and the results are given in Table 3, along with the
experimentally ctserved values and other calculated results.
In Figure 2, 1 simulated photoelectrcn spectrum is shown
where the profiles are generated ty qaussians of ccrstant
halfwidtl (FWHM=.r'eV) weighted by the spindegeneracy of
the resultant ionic states. The inset is a tracing of the
ESCA result (SiEcbabn et al., 1569). The prevailing feature
is a shift to higher energies, particularly for some of the
triplet states relative tc the singlets. In order to retain
a total energy which is independent of the model
Hamiltonian, the occupations again were kept fixed in this
calculation. The atomic orbitals, also independent of the
66
Table 3. Calculated and experimental ionization potentials
for nitric oxide (X2n).
Ion a 1 c d this Q f
state MO expt expt exFp calc calc calc

in C1s' 544.C 564.2
3n Cls 543.3 565.4
In1 N1s' 4 11 42r.1
3n NIs 410.3 422.1
in 3C' 43.8 51.2
an 36 40 .6 47.6
In 4 23.3 28.0 21.0
3n 44 21.7 29.9 20.0
Ain 59 18.5 18.3 18.3 18.6 19.4 17.8
3n 56 16.7 16.5 17.2 16.1 17.0 16.9
i + 1ny' 17.8 18.3
I 1nx' 19.3 18.9 15.8 17.7 17.8
i 18.4 17.3 17.0 17.2
35 17.6 16.6 17.0 17.C
3 1Inx 16.8 15.7 21.9 16.0 1 1
a35+ Iny 14 15.7 14.0 19.4 15.0 14.3
Xiy + 2 .y 10 9.3 9.3 15.8 9.3 12.0

a
Via KooFPans' theorem.
t
ESCA, Sieqbahr. et al. (1969). (Vertical)
C
Turner et al (1970). (Adiatatic)
d
Collins and Natalis (IS19). (Adiatatic)
e
LefebvreBricn and Moser (1966). (Adiabatic)
f
Thulstrut anc Oirn (1972). (AdiabaFtic)
Piqure 2. Cicula+ef and experimental FhctoF)Actron
spEctra for ritric oxide (X2rC) 'onvoltiticn done with
aaussians with FtrHM;=.5eV. Inset fEom ESCA (Siegbahn, et
al.. 1965).
NITRIC OXIDE
AJ
1 1 11 I 1 1 1 I Ii 1 1
40.0 30.0 20.0 10.0
69
model Hafiltonian, are illustrated in Figures 3 and 4 by
"hiddenline" picts (McTntosh, 1975) in +he yTplane for
nitrogen and oxygen respectively. The inclusion of virtual
orbitals in thc model Eamiltonian tends tc be a sensitive
issue as can te the charge transfer, but unlike the charge
transfer problem, it has no direct effect on the total
energy or the atomic orbitals, provided the occupations are
restricted to zero and unity.
At this point, cnp might begin to have some doubts about
the various apprcximations used ir cur treatment, or the use
of a single nensymmetric spinpolarized valencetbnd
configuration. One must reserve judgement, however, until
each of these approximations has teen carefully examined on
a variety of systems, for one can not help but single out
the cmdel Hamiltcnian as still a prime target of scrutiny.
one might also te suspect of KccFmans' approximation for
this system. A spinprojected unrestricted HartreeFockz
calculation would help in this regard.
let us now lock at what should be a more promising side
of this story. The total energy that we obtained at the
equilibrium geometry (uith fixed occupations) was about 6 eV
2A spirpolarized Xalpha calculation of Connolly et al.
(1973) is in reasonable agreement with experiment.
'Using the same basis and GrandCanonical HartreeFock.
Figure 3. "Fljddonljn&' Flots of the atoric crtitals
on the nitroqEn atcu. yzrane (x=C.O).

ii:
~liv,"
__; : ; .1
*:~ ., I" 
I I
~~""' '' r~ ::~T~ .
'1
"'
i!
Figure 4. tHildenlinel Ficts of the atomic ortitals
cn the olpqOn atcm. yzplane fx=C.P).
73
 l II;
l!1;II I1"
 ..'
p
;:. .
';k
r ~Y ~

_.: i
=; ~ ~~ ~~~~ ~  r
i "
i: ...: ;
1  ~ 
74
lower than that of the separated HartreeFock atoms.1 The
experimental cissociaticn energy is 6.5 eV. This is either
forttitcts or very encouraging. A full potential energy
curve is certainly in order at this point. For the rcst
rapidly ccnvercent procedure, cre starts the potential
energy curve it the separatedatom limit where one is
guaranteed the proper limit (by ccnstructicn) and proceeds
alcnq the curve toward smaller interatomic distances, using
the results cf each calculation as a starting point for the
next. This procedure is particularly important for charge
consistent calculations (which is not the case here,
however). Thq results cf these calculations are presented
in Figure 5 and summarized in Table 4. The spectrcsccpic
constants are determined by a leastsquares fit tc the
HultertFirsctfelire (1941) modified rorse potential
function and interpclaticn is generated by a cutic splire.t
The agreement with experiment is quite remarkable so much,
in fact, that one is a little reluctant to stake any claims
as tc its veritaLility.
It appears 'hat we have witnessed both extremes cf cur
earlier expectations. Perhaps, though, the latter success
for the potential energy curve for NO (X2n) can offer
recourse to a resolution of the apparent failure to describe
IProqram written ty Nelson H. F. Beebe, (Quantum Theory
Project, Eniversity of Flcrida, Gairesville, Florida).
Figure S. uvalcula+ and experimental (Morse)
pctfntial enerqy curves fcr NC(X~n).
76
NITRIC OXIDE
0.0 1.0 2.0D 3.0 '. 5.0
R (BOHRI
7,
cc D1.I
a:
Lh
F
77
able 4. Calculated and experimental spectrcscopic
constants for nitric cxid (X2n).
a this
Ccnstent expt calc calc
r (Ao) 1.15 1.10 1.25
e
1
(cm ) 1904 1993 1731
C
1
w x (ca ) 14 20 24
e e
1
B (cm ) 1.70 1.86 1.45
E
1
0( (cm ) C.018 0.C32 0.023
e
D (eV) 6.5 e.C 3.8
e

a
Herzberc (19c0).
b
ThulstruF and Ohrn (1972), uses sate spline fit
as in present wcrk.
the ionizaticn process. That is, one riqht anticipate
similar success in commuting the potential energy curves for
the variccs states of NC+ for a direct calculation cf the
ionizaticr potentials. He shall end the treatment of nitric
oxide on this hopeful note since our real aim here is
sceewhat different.
78
3. Nickel (lOC) Surface
Fuch has been said in these energyconscious times atcut
the importance of tte study cf surfaces and of chemisorption
on them in the field of catalysis, and the recent growth of
interest in this field has spawned significant advances in
toth theory and experiment. iltrahigh vacuum techniques
coupled %ith an endless variety cf optical, electron and
other spectroscopic measurements provide the surface
scientist with the tools to help understand the nature cf
surfaces and the electronic structure of surfaceadscrtate
ccmplexes.i Or the theoretical side, various approaches stem
from both solidsta'e band theory and theories generally
applied to molecular complexes; however, we are still far
from possessing a reliable computational method for a
'See, e.g. Physics Today, 28(4), April 1975; issue
devoted tc surface physics.
79
satisfactory description of the chemisorption bond.1 One
needs a metbol as simple as extended HUckel theory
(Wolfsbera and Helmholtz, 1952; Anderson and Hoffmann, 1974)
which is selfconsistent and contains no empirical
parameters.
The approach we take here is, of course, a localized one
with respect to the rcle of the surface in the chemisortricn
bcnd. In order to avoid the effects of cluster boundaries,
we shall compose a finite atomcluster model built up from
atoms which retain their "bulk", "surface" and "adscrbate"
identities. Fich atcn in the cluster will still "feel" the
influence of at least its nearestneiqhtor environment, even
though these noiqhbcrs [ay not he explicitly in the cluster.
This, we feel, is a very important feature to be
incorporated in smallcluster simulations of extended
systems. One would like to have a "surface" cluster which
does not have surface of its own, so to speak. We shall
construct our model in three steps.
The first step is tc characterize the hulk, and by this
we mean a selfconsistent description of an atom in the
metal. In the case of nickel, we have a facecenteredcutic
(FCC) lattice which frcn the viewpoint of a single atom is a
ISee Bullett and Cohen (177a) and references therein for
a short synopsis of current techniques.
80
nearestneighbor shell of 12 other atoms in a cutcctahedral
arrangement (see top structure in Fiqure 6), each of which
sees the same environment about it, and so on, to complete
the fully extended tulk arrangement. As a first
approximation, we will consider nearestneighbor
interactions cnly. The selfconsistent calculation proceeds
as follows. We start with a free atom calculationi using
the hybrid sirqlEdcublezeta basis given in Tatle 5 and a
spherical 4terr model potential expression with the
exponents in able 6. We then take 13 of these atoms with
their freeatcm crtitals, energies and model potentials, and
arrange *hem as in Figure 6 (op).2 The experimental lattice
constant of 4.7 a.u. is assumed throughout. The next
calculation is cr the central atom in this cluster with the
12freeatom environment taken fully intc account. Now
comes the first impcrtart step toward selfconsistency. We
take the results of the central atom and place them at the
12 neightcr sites and repeat the central atom calculation
with this modified ervicrnment. This process is repeated
until the central atom and its neighbors "coincide". The
eiqenvalue spectrum or this selfconsistent "atominmetal"
is shown in Fiqure 7, ccopared tc that of the free atom.
Cne cculd, in principle, extend this into a hand calculation
'An unrestricted GrandCancrical HartreeFock calculation
fcr the 3F state (de configuration).
zFerroiaqnetic couplirq (triplet pairing) of spins.
Fiqure f. Cluster models for the bulk, surface and
fourfold hcle lscroticn site.
82
83
Table 5. Easis set for nickel.
a
Type exponent

is 21.3410
2s 1 .1C08
2p 12.0478
3s 4.9860
3p 4.6940
3d 6.7055
3d 2.8738
Us 2.0771
4s 1.1389
a
Mixed sirqle ard doublezeta basis cf
Roetti and Clementi (1974).
able 6. ?odel potential exponents
for nickel.

a
lype exponent
is 1.5193
Is 3.0366
Is 6.0732
Is 12.1460
a
Eventempered set (0.5,1.C,2.0,4.0)
scaled hy Z** (/3) .
Figure 7. Calculated Fhctoelectron spectra of
Ni(atom), Ni(netal), and Ni(surface). (ranqe O140eV).
85
NI RTOM
NI METAL
NI (100)
SURFACE
86
just as .ith the renormalizedatom approach (Hodges et al.,
1972) and in 9 certainly more elegant, if not superior,
fashion. Such a venture has rct teen undertaken to date,
however.
The next step is to characterize the nickel (100) surface.
Analoqous to a hulk atom, a (100)surface atom has only E
nearest neightcrs (see riddle structure in Figure 6), if we
ignore arv distortions for the moment. As a first
approximation, we can consider the top layer as teirg the
only layer which is different frc these in the bulk. We
shall also, as in the bulk case, consider only nearest
neighbor interacticns. From transformdeconvcluted LEED
(low energy electron difracticr) experiments (Landman and
Adams, 1974) the (1"0)surface layer spacing is within 11 of
the bulk value for Ni (FCC), so no distortions ir the
geometry reed to be taken into account. Now, since fcur of
the atoms in our model cluster are in a "tulk" layer, we
hold them fixed in their "tulk" state (same orbitals,
energies, model potentials, etc.). The other four are
surface atoms identical to the central atcm. We will thus
proceed as before with a calculation for the central atom
taking into account its full nearestneighbor environment
(the other four surface atoms starting out in their bulk
states), and then iterate, by replacing the four neighboring
surface atoms with the results cf +he central atom
87
calculation, urtil selfccnsistercy is reached. For
practical reasons yet to be worked cut, the calculation for
the nickel surface did not converge. despite various
damping techniques emFlcyed, only four iterations could be
obtained before strange things happened. The eigenvalue
spectrum for the fourth iteration is displayed in Figure 7
alcng with that of the free atom and the metal atom results.
It was also recessary to hcld the rcdel potential fixed to
its free atcm value, and for consistency, the hulk
calculation was redone using the same model potential (the
differences were minor, however). The shift to higher
finding energies is somewhat unexpected and is perhaps only
an artifact of the converaence ancoaly, but we shall take
what we can get and proceed.
The last step is the construction of a model cluster
representing the most likely site for adsorption. Based on
theoretical and experimental studies of H, 0, and CC on
Ni(100) surface (Bullett, 1977b), the preferred site for
adsorpticr is most likely a fourfold hole site. This was
also the site chosen for a previous calculation of nitric
oxide on nickel by Batra and Brundle (1976). Nearest
neiqhbor interactions for this site would involve 5 nickel
atoms four "surface" atoms and one "hulk" atom (see bottom
structure in Fiqure 6). We are now ready to bring the
adsorhate intc the picture.
P8
. Nitric Oxide cn Nickel(102) Surface
The catalytic reduction of nitric oxide (by hydrogen)
over noble metal surfaces is a reasonably wellknown process
(K~bylinski and Taylor, 197) It is, in fact, the
underlying process of the catalytic converters installed in
neter emissicrccntrolled automohiles (did you ever wonder
where that "ammonia sell" came frcm?). We are not quite
ready to tackle the actual chemistry that takes place on
such a surface, but we can try to entertain a couple of
questions that some recentt experiments have left unanswered.
Nickel is nct a notle netal, but it is a known catalytic
aqert fcr mary reactions involving the dissociation of a
nitrcqencxyqen bond, such as the reduction (by hydrogen) of
nitre ccnrounds to amines (icrriscr and Boyd, 1966) which is
probably the most important synthetic route in aromatic
chemistry (the introduction of ether groups into arcmatic
89
rings via a diazonium qrcup which is readily obtained from
primary amines), since nitro compounds are easily prepared
by direct nitration.
So why do we want tc lcck at nitric oxide on a
nickel(10C) surface? Cne might think of this as a prototype
system fcr the tyre cf reactions just mentioned, but more
specifically, recent Xray (XPS) and ultraviolet (UES)
studies cf nitric oxide and nitrogen dioxide interactions
with nickel (Erundle, 1976), and infrared (IP) data for
nitric oxide on nickel (Blyholder and Allen, 1965) have
posed a couple of interesting questions. There is
reasonable evidence tc suggest that nitrogen dioxide
dissociates or nickel even at very low temperatures (80K)
leaving ritric oxide and atomic oxygen adsorbed on the
surface. There is not enough data, though, to say that the
direct reaction of nitric oxide hith nickel produces the
same species on the surface (same adsorption site, same
orientation, same electronic state, etc.). Upon warming (to
3CCOK), adsorbed nieric oxide slowly dissociates into
nitride and oxidelike species: however, the former
reaction is accompanied by the less of atomic nitrogen.
Subsequent reaction at this temperature with nitrogen
dioxide and nitric cxide, respectively, produce very similar
XES and CES spectra, with loss of nitrogen, and, in the
latter case, the appearance of a new1 weakly bound state of
90
nitric oxide easily described cn heating in vacuum). 'he
UPS spectra under these conditions were uninterpretable.
there is good reason to believe that nitric oxide is mcst
strongly tound with nitrcqen crierted toward the surface in
a fourfold hole site perpendicular tc the surface 1lare.
However, could it he that nitrogen dioxide initially
condenses with cxyqen oriented toward the surface (since
both ends have exposed oxygen atcms) and then dissociates tc
leave oxygen tonded nitric oxide cr the surface which later
dissociates (upon heating) to give off nitrogen? Could the
weakly adsorted state observed during extentsive reaction
with nitric oxide also be a reversed orientaticn of nitric
oxide which tier also dissociates tc yield nitrogen? In the
IR spectrum of nitric oxide on nickel a weak unassigned band
occurs which has been attributed to a different adsorption
site. Eut perhaps this too can he accounted for by this
alternate orientation. Thus, if ore could calculate the
binding energies for nitric oxide on nickel in bcth
orientations and predict the observed shifts in the UFS and
XPS spectra, it would certainly lend support to this
hypothesis. A more ambitious check would be to compute a
potential energy curve for the nitrogenoxygen stretch in
this alternate orientation and predict the observed
unassigned frequency in the IR spectrum.
INew with respec+ to the original UPS and XPS spectra of
directly adsorbed nitric oxide before further reaction was
allowed tc take place.
91
Cur first calculation was on a cluster of 5 nickel atoms
arranged as in Figure 7 bottomm) with the nitrogen atom
oriented toward the surface and the spins
antiferrcnaqnetically aligned with the ferrcmagnetic surface
(one might think of the entire surface cluster as a single
Hund'srule atom). The nickel lattice constant was left
unaltered at U.7 a.u., the nitrogennickel distance was set
at 3.7 a.u.1 (for all four surface atoms), and the ritrogen
oxygen cistarce was kept at its equilibrium molecular
distance cf 2.17 a.u. As mentioned in the previous section,
the orbitals ani modl potentials for the nickel atoms were
fixed at the tulk ard surface values; hciever, the
correspcrdinq enerqies were computed correctly for their
environment. Ihis is only a first approximation but the
next level of approximation would require (without
repreqraTuinq a+ this point) a much larger cluster in order
tc include a proper environment for the surface rickel atcms
(with proper stcichicnetry). Fcr a coordinate1 site (on
top of a nickel atom) this could have teen done with the
nineatoa cluster in Fiqure 7 (middle). The total density
for nitric oxide contributionss from nitrogen and oxygen
atomic ortitals) is shown in Figuro 8 (bottom) compared to
molecular nitric oxide (tcp). the plet plane in this figure
is 0.3 a.u. above the yzplare to suppress the Is
ITaken from nickel oxide lattice.
Fiqure 8. Total density ccntcurs for nitric oxide
(top) ard nitric cxide on nickel(100) surface (bottom).
(x=0.3 a.u.)