Group Title: projected Hamiltonian approach to polyatomic systems
Title: A Projected Hamiltonian approach to polyatomic systems
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Title: A Projected Hamiltonian approach to polyatomic systems
Physical Description: viii, 115 leaves : ill. ; 28 cm.
Language: English
Creator: Smith, Jack A., 1949-
Publication Date: 1978
Copyright Date: 1978
 Subjects
Subject: Binding energy   ( lcsh )
Hartree-Fock approximation   ( lcsh )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 111-114.
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Jack A. Smith.
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Bibliographic ID: UF00099255
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000065650
oclc - 04361963
notis - AAH0864

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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 Per-Olov 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 74-2656C), and the National Science

Foundaticr (NSF CHE74-01948, SMI76-23036).


-iii-


1

















TABLE OF CCNTFNTS


ACKNCWLEEGEMENTS ...................................... iii

LIST OF TABLES ........................................ V

LIST OF FIGURES ....................................... vi

ABSTRACT .............................................. vii

CHAPTER

I. INTRCDUCTION ................................. 1

II. THE EFFECTIVE CNF-ELECTFCN CPEBATOF
AND THE PARTITIONING OF ITS EIGENSFACE ....... 5

III. THE ENEFGY-DFFENCFNT PSEUFOPOTENTIAL ......... 19

IV. A MOEEL APPRCXIMATICN TC THE EFFECTIVE
CNE-ELECTECN INTEFATCMIC CCULCMB POTENTIAL ... 28

V. AN ATPRCXIMATE NCNLOCAL INTERATCIIC
EXCHANGE FCTENTIAL ........................... 37

VI. A MODEL HAMILTCNIAN AND AN APERCXTIATE
ELECTRCO FSOEAGATOR .............. .............. 39

VII. THF GRAND-CANCNICAL HARTREE-FCCK SCHEME ...... 45

VIII. A SEIF-CCNSISTENT 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. "Hidden-line" plots of the atomic crbitals cr the
nitrccen atc ......................................... 71

4. "Hidden-linp" 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 four-fold
hole adsorption site................................ 82

7. Calculated p-ctoelectrcn 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 "atcm-in-

molecule" nature. Within an independent-particle 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 second-order 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 one-electron 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

Hartree-Fcck 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), valence-bond (VE) (Gerratt, 197U), atom-in-

molecule (AIM) (joffitt, 1951), and similar building-block

(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 one-center 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 tuo-electron









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 m-thods 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 cne-electron

operator; tc onplcy analytic techniques enabling proper

treatment of the exchange interacticns; to use suitable

basis sets, such as Slater-type orbitals (STO's) of at least

"double-zeta" (E2) quality with polarization functions (DZP)

when needed; to correctly compute and retain all one-center

inteqrals and all two-center one-electron integrals; to

reduce all multicenter two-electrcn integrals to two-center









one-electron integrals by exploiting the !ccali2ed picture,

but %ithcut resorting to empiricism or neglect of any

differential overlap; and to establish atcmic valency as a

tetter-definec ard wcrkatle ccncep*.



In the next chapter the effective one-electron

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 one-electron 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 Hartree-Fock approximation is described

which allows for the definiticr of a self-consistent

potential from a density ccrrespcnding to a statistical

ensemble with specified occupation numbers. The prchlem of

spin-orbital 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 CNE-FIFCTRCN 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, A-B. 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 A-B.



The average value of the ncnrelativistic many-electron

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 ) (IT-1)
i i ii ii i i i i

where the a's and t's are field operators corresponding tc

spin-ortitals assigned tc ators A and B, respectively, and

the summations have been restricted accordingly. let us

assume that the spin-orbitals assigned to ator A are









expanded in a tasis centered cr A and the spin-ortitals

assigned tc atom B are likewise centered on B. Such a total

energy functional leads to the effective one-electron

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 (II-2)
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) (11-3)
1 i i i 12 i

and

A (A) A A -1 A
K 4(1) = q <4 Ir It> (1) (11-4)
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 sFin-crtital occupation numbers. Analogous

definiticrs for the Coulomb and exchange operators on atcir

are assumed. The sFir-crbitals are no- restricted to be

eioensoluticns -c a pseudoeigenvalue problem,

F 4 (1) I S (1), (11-5)


althcuqh they lust be solved for self-consistently since the

operator deperds on their thrcuqh the Coulomb and exchange

oTerators.









At this point we have dore nothing but invoke the

Hartree-Fcck approximation with the only twist being the

localized restriction on the spin-crbitals. The canonical

Hartree-Fcck scheme would require that the spin-ortitals be

symmetry-adapted, 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 well-localized.

We shall use such localized equivalent orbitals in cur

treatment and restrict, instead, our spin-orbitals 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+om-in-molecule partitioning

of the pseiidoeiqenvalue problem in (11-5). To effect the

partitiorinq, let us lock at the diatomic molecule A-B at

very large interatomic separation where the fifth and sixth

terms in (TI-1) become negligible and the eigenfuncticns of

F in (II-2) approach the atcmic cancnical Hartree-Fock 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] (II-6)
i 1A i










E B 2 R -1 E E B
F t rCi = (-1/2 V 2 r + J [C] K [0] )} [0] (11-7)
i 1i i

where the [01 derotes this separated-atcm case (dropping the

reference to clectrcn 1 on the cperatcrs and the functions).

It should be cnphasized t-at 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 (II-2) fcr all

of its eicenscluticrs, we seek cnly those scluticns which we

associate with atom A, while the solutions of (11-7) for

atom B are assumed tc suffice, for the moment, as the

solutions of (11-2) associated with atom E. that is, we

wish to solve *he pseudceiqenvalue problem

A 2 A A A
F ( [ 1] = 1-1/2 5 2 r + J [1] K [ 1
i IA

B -1 E B A
-Z r + J fC] K [f] ) 4 [1] (11-8)
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 (11-2). The [11 here

denotes the first iterate of this higher-level 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 (IT-9)
i i

must be satisfied. It is sufficient, though, that they be

crthcqcral,

A B
= C, v 1, (iI-I 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 = 1-1/2 i 2 r + J [1] K [ 1]
1 1B

A -1 A A B
Z r + 31 [C] K [0] } 4 [1] (I1-11)
1A j

subject tc the ccnstrairts

A B
< 0 1 4) [ 11> = c, v j,i (ii-12)
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 n-1],
i i

E E
4 [nl --> 4 rn-1], (II-13)
i 1

and we satisfy tte crtlcqcnality conditions

A B
<4 rn]i r > = C, V ij. (II-1)
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

t-inc. 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
= (1-P ) 0 (IT-15)
B i

where

+ (E) 1 E
P = F = 1 > E E i i 1

is the prciection operator which projects cut the "B-Fart"

of the furccticrs on A. Condition (II-1Q) in terms of the





-12-


unconstrained fi actions (dropping row the [n]-notaticn at

self-consistercy) becomes automatically satisfied,

A B A + B
<4 IFI| > = < I (1-P ) 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 (11-17)

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(1-F ) 4 = 8 (1-F ) 4 (11-18)
E i i B i

and the secular protlei cwuld then become

_A + A
1<4 I 1-P ) IF-G)(1-P ) It > = 0. (11-19)
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 (1-P ) (F- )(1-P )1| >1 = C (11-20)
i A A i





-13-


with analcqous definitions fcr tho unconstrained functions

on atom E and for *h- projection operator associated with

the "A-part" cf A-B.



If we rewrite equaiicns (IT-19) and (TT-20) with the

unprotected operator F separated out, we get (Weeks et a].,

1969)

A -_
I1 = C (11-21)
i A j

and

B E
I< | IF+V -814 >1 = C (11-22)
i B 1

where the pseudcpotentials are qiver by

V = -(P F-P ) (2-r ) [F,P ] (11-23)
A D B B P

and

V = P F-SP ) (2-P ) [F,P ]. (11-24)
P A A A A

It should be noted that the second term in (11-23) and

(11-24) will vanish if the operator F ccmmutes with the

prciecticr operators, that is,

rF,P 1 = rF,F 1 = C. (II-25)
A 9

Furthermore, if the "projection" operators are truly

idempotert, then these pseudccotertials reduce to

V = -IF-6)P (IT-26)
A E

and


v = -)P .
E A


(II-27)










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 independent-particle 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 (II-28)
A i i i

_E E B
(F+V ) 4 = F 4 (II-29)
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 single-center

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 spin-orbital could

be unambiguously assigned to a particular atom and, rcre

importantly, expanded in a single-center 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 spin-orbitals, 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, (11-30)
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
(1-P ) i (1-P ) (IT-31)
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 (11-32)
i j,k k i k j k

If we now invoke the Mulliken apprcximation (1949) for the

differential overlap,

*
4 = <# I4 >( + 4 )/2, (1-33)
i ti 1 1 i i 1

then (1I-31) 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 (IT-34)
i i i

where

A (B) A B 2 -1/2 A
S' = (1 <4 (4 > ) 4 (11-35)
i i ii i

and

A A
<4 'I '> = 1. (IT-36)
i i

Thus, the rulliken approximation leads directly tc a fcra cf

the tctal density ir terms of the sinqle-center

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 '. (11-37)
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 single-center 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

(11-19), that would te associated with atom A in the

triatomic molecule A-B-E' is

_A + + A
I<( I (1-P ) (1-P ) (F-b) (1-F ) (1-P ) I >1 = 0 (TT-38)
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) (IT-39)
E B' j k 1 k k i

where

E E'
1 0. (II-40)
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 second-order 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 ENEEGY-CEPFNDENT F5EUDOPOTENTIAL








In the previous chapter, ie have essentially defined an

effective atom-in-mclecule one-electron Basiltonian. Its

associated eicenvalue prchlem has the fcrm us that


Q FQ 4 = G t (III-1)

where F is the polyatcmic analog of (11-2) and Q is a

product cf prciecticn cperatcrs,

C = (1-P ) (1-P ) (1-F ) ..., ( II-2)
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 + ..., (III-3)
B B' D P' B B"

and retain only terms up to second-order in differential

overlap Iseccnd-crder 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) (III-4)
i i i 1 i i j1

we get tbe qereral form for Q


-19-





-20-


C = 1 P (TI-5)

where

(E) (B') B B' 2 U B
P = (1 <4 14 > )1j ><4 1. (IIT-6)
i i-i i 1 i i

The first sur in (ITI-6) 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 > (111-8)
i jii i 1

being an overlap correction to the "projection" manifold.

In the limit of zero-overlap, 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 (11-19), namely





-21-


1< I 11-P) (F-8) (1-F) I >1 = C (III-9)
i 1

or in terns of a pseudo Fctential

II = n (III-10)
i 1

where, analoqcus to (11-23),

V = (FF-E( ) (2-P) [F,P]. (III-11)

In order to simplify the form of V, as in (II-26), 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. (III-12)
i i i i

We must recall ttat the eigenfurcticns in (T11-12) are not

eiqenfuncticns cf F but rather

B 0D B
(F+V ) 4 = 6 4 (IIT-13)
i i i

with the corresponding pseudopctential associated with the

environment of atom B. Rearranging (III-13) gives

S B E B E
F 4 = t V 4 (111-14)
i ii i

which through first-order in perturtation theory becomes

B B E
F = G < (III-15)
i i 1

where

E E B E P
S = 8 <4 IV 14 >. (TII-16)
i i i i

Using thij now in (II1-12) qives





-22-


(13)_E E e
FP = 6y I > i i i i i

and since F is a self-adjcint oFeratcr, (111-15) also grants

FF = FF (III-18)

causing the commutator in (III-11) to vanish, leaving us with

V = IF-g) (2P-PP). (111-19)

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 (IIT-4) 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 > (III-22)
i i jti i i j

Sutstitutinq (111-17) and (III-21) into (TIT-19) we obtain

(E) _E E E F E
V = (-E )(2-y ) y I ><0 I (ITI-23)
i i i i i i

which in the limit of an orthonormal "projecticnn" marifcld

reduces tc the lFillips-Kleinmar (1959) pseudopotential

IE) E E B
V = (&-_ )14 >< I. (ITI-24)
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 (IrT-10).

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 (111-25)
i k=1 k ki

cr in matrix notation

4 = u C (III-26)

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 (III-27)
i k k ki

or in matrix notation

E E B
4 = u C (III-28)

If we now define the matrices

B
S = , (III-29)
kl k 1

(B) _B B E B*
P = (2- )Y C C (III-30)
kl i i i ki li


and










(B)_B _B B B B*
R = % (2- )y C C (III-31)
kl i i i i ki li

then

V =
kl k 1

t +
= E (SES ) (SES ) (IT-32)
kl kl

Thus, asile from the energy dependence of V, the secular

prctlem (III-10) reduces tc finding the unitary matrix C

such that

+
C (F+V)C = & (III-33)

where

F = (TIT-30)
kl k 1

and & is now a diaqcnal matrix.



As already stated, the only remaining complication is the

explicit energy dependence of the pseudopotential (III-32).

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 one-dimensicnal filtert

space, are also inherertly rcn-Hermitear 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 (111-33) in terms of a modified Fock matrix, that

15,

+
C FC = 6 (III-35)

where


F = F+SP-R, (IIT-36)

t
P = SES (11I-37)

and

t
B = SES (ITI-38)

Then we note that from (III-35)


PC = CC. (11-39)

Noh putting (III-36) back into (TIT-35) we have

4 + +
C FC+C PCG-C BC = (III-40)

which upon substitution of (TTI-39) becomes

+ 4 +
C FC+C PFC-C BC = (III-41)

or


C (FEF-R)C = (III-42)

So ccmparinq this with (11T-35) we have that


F = F+1F-B (1II-43)

which upFc rearranqinq qives

-1
F = (1-P) (F-R) (IJI-44)





-26-


If we now expand the inverse, we generate the series


F = F-F+PF-PB+PPF-... (III-45)

frcm which, alcnq with (III-36) and (II-43), we gather that


SP = iF = EF-EP+FPF-... (IIT-46)

which is similar tc a perturbation expansion. Since the

pseudopotential matrices (II-37,II-38) are already seccrd-

order in cverlap, this series should LP rapidly convergent.

In fact, we shill take crly the first two terms of (III-46)

in our approximation. The non-Hermiticity is also apparent

in this expansion form, and so we choose the approximate

Hermitean form for (III-46)


SP = (EF-PF+FP-RP)/2. (1II-47)

Our modified Fock matrix then becces


F = P-J+(FF-BP+PF-P) /2. (TTI-48)



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 self-consistent manner as described in the previous

chapter. All aspects of the Hartree-Fock scheme (cr any

generalization of this independent-particle 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 one-center expansion of





-27-


the eiqenfuncticrs; however, we have not yet used this to

its full advan+aqe. Although we have eliminated all three-

and four-center two-electron irtegrals and many two-center

inteqrals, we have only replaced them by just as many cre-

and two-center 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
CNE-ELECTRCN INTERATOMIC CCUICMB PCTFNTIAL








If one can substantially reduce the rubber of multicenter

twc-eloctron 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 two-center integrals

by restricting tie eiqerfuncticns to cne-center 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 two-center

Cculcmb integrals. This shall be done ty replacing the

effective one-electron cperatcr, for the two-electron

Coulomb interaction, by a rcdel approximation.



The irteqrals which we wish tc approximate are of the

form

E
<4 l1J K > (IV-1)
i i


-28-









which is the Coulombic interaction of the i-th spin-orbital

with all the ctter spir-crhitals in the polyatomic system

except thcse on the same atomic center (A). The effective

one-electron operatcr is given more explicitly by

E (U) F E -1 P
J = q <4 r d1 > (IV-2)
1 i 1 i 12 i

where the q's are spin-crtital 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 two-electron integrals cf the type

E -1 U
. (IV-3)
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 (IV-2)

by the nuclear coordinates of atom P, that is,

E |B) E E -1 F
J = q <4 It >. (IV-5)
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 1-Y (r ) Ir (IV-6)
1B 1P

where X is a screening function with the asymptotic behavior


X (0) = 1 (IV-7)

and

E
X (oo) = 0. (IV-8)

Such a screening function is inherent to the Thomas-Fermi

model (Thomas, 1928; Fermi, 1928) of the atom. A

sionificart feature cf the Thoras-Fermi screening function

is that it is universal for all neutral atoms with respect

to the dinensicnless variable





-31-


2 E -1/3
x = r (3-n/8) Z ] (IV-9)
1E 1B

in terms of the nuclear charge. The distance r is assumed

to be given in Bohrs. Pctentials of the form (IV-6) are

certainly nothing r-w 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 ) (IV-10)
1B t t t t t t t ID

where

p -- n-1
X (n,l,m,a:r ) = r exp(-ax )Y (6 ,a ) (IV-11)
t 1F 1F 1E Im 18 1B

and where t-e 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 (IV-6) tc its true form in (IV-5)

alona with the constraint in (TV-7). We have merely

expanded the operator J in a basis of Slater-type 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 (IV-6) for (IV-5) in the one-center Coulomb









inteqrals cccurrina or atom B and matching their values,

that is,

B _- P B B P
<4 IJ 14 > = < IJ II t > (IV-12)
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 I|t > 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 > (IV-13)
1 1 i i 12 i i

%here the Coulcrb integrals on the righthand side cf the

equation are cne-center inteqrals which we properly compute

and retain. he have as many equaticns like (IV-13) 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 (TV-1C), though, are usually much less

than that (due tc its relatively smooth behavior), sc scme

type of weightedd) least-squares fit procedure should he

adequate. One other condition, perhaps even more important,

that can be satisfied is the +ctal twc-center nuclear

attracticr experienced by that atcm, qivon by

A (B) B F -1 B A -->
Z q <4 Ir I > = Z J (r ). (IV-14)
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 (IV-13), 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 >. (IV-15)
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 (IV-16)

where

E
A = A (IV-17)
t t

B E B -9 -1 B
T = 1 <0 IX (r )r !( >, (IV-18)
it i t IP 1B i

E B -> -1
T = N X r )r (IV-19)
M+1,t t AE AB


T = 1 if 1 = C,
M+2,t t

E
= 0 if 1 > C, (IV-20)
t

E B -1 E
G = N


i i 12 i

(B) E E -1 BE
q < 4 Ir 1i 4 >, (IV-21)
i i i i 12 i 1

B -1 (B) B B -1 B
G = N r q , (1V-22)
M+1 AB 1 1 j 1A i

and


G = 1.


(IV-23)





-311 -


In general M+2 is larger than the number of terms needed in

(IV-10), and so we have an overdetermined set. If M+2 and N

were equal, then T would he square and

-1
A = T G, (IV-24)

provided that T is non-singular (no linear dependencies).

However, in the general case, one can write the normal set

of equations IIV-16) as

t
(T vT)A = IT w)G (IV-25)

where w is an arbitrary diagonal matrix whose elements act

as weiqhts +c the original M+2 equations (IV-16). The

choice of a urit matrix cwuld ccrrespcnd to the usual linear

least-squares fit procedure; nevertheless, the option of

giving more importance to certain conditions such as the

external ruclear attraction (IV-11,15) 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. (IV-26)



The exponents in (TV-lO) could be determined once and for

all by a mere fit to the universal Thomas-Ferri screening

functior nenticned earlier, hut this would build in the

deficiencies of the Thomas-Permi 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 Thoras-Fermi equations with the

following trial function for the screening function:

'x IX -a"x 2
X (x) = (A'e A"e ) (IV-27)

His calcula-icn yielded the exponents

a' = C.178255C and

a" = 1.759339 (IV-28)

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 (TV-10):

a = 2a' = 0.3565118,
1
a = 2' + a" = 1.937594, and
2
a = 2a" = 3.518678. (IV-29)
3

These values perc derived for spherically symmetric neutral

atoms in the Thomas-Fermi model with somewhat modified

boundary conditions iallied by the particular choice of

(IV-27). In crier to allow for corrections to the Thoras-

Fermi model, radial and anqular distortions, and deviations

from neutrality that would exist for an "atom-in-a-

molecule", some additional flexibility (other than the

variation of the linear coefficients) might be necessary.

Auqmentinq expression (IV-10) 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 iell-rourdel, 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

well-determined lirear ccefficierts aculd he sufficient.















CHAPTER V

AN AFFECXIM'AT NCNLCCAL INTEPATOMIC
EXCIHANGF PCTENTIAL








In the last chapter, we substantially reduced the number

of thc-center two-electron 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 one-electron operator, hut

this time the operator is ncnlccal and requires a somewhat

different approach.



The tc-certer excharqe integrals are of the form


<4 K 10 > (V-1)
i 1 i

where K is the effective nonlccal one-electron operator

subscriptt 1 dropped from here on) for the exchange

interaction with all the other spin-orbitals (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). (V-2)
i j1 j 12 i 1





-38-


Let us consider the asymptotic behaviour of (V-1) when the

average irterelpctrcnic distance approaches the internuclear

distance, that is, %hen

E -1 (1) B B E
< 1K 14 > = r q <4 104 > (V-3)
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 secend-crder in diatcmic cverlap, and so it is felt

that retaining K in its asymptotic form (V-4), which is

second-order 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 two-electron

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 >> ] (VI-1)
i 1 + i j F

where

a = <4 I~ > and
i i cp


a = <4 14 > (VI-2)
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 >, (VI-3)
i 1 + i + ii i j


-39-





-40-


defined in terms of field operators and a spin-crbital

tasis. The iany-electrcn Hariltcnian, in second quantized

fcrr, is

+ + +
H = h a a 1/2 2 (klij)a a a a (VI-4)
ii ii i i ijkl i k 1 j

where

h = < | h it > (VI-5)
ii i 1 1

and

-1
(kl(ij) = <$ 4 Ir I4 > (VI-6)
k i 12 1 j

are the usual cre- and two-electrcn integrals. Equation

(VI-1) can be iterated to yield

-1 -2
G (E) = S t F F + . (VI-7)
ii ii ii

where

+
F = (VI-8)
ii i i +

is the first acrent and sc on. Substituting fhe Hamiltonian

(VI-4) into (VI-8), the first moment becomes

+
F = h + r (iilkl)-(illk ) ]
(VT-9)
ii ii kI k 1

which has the same form as the effective one-electron

Hariltonian as originally presented by Fock (1932) and will

hencefortl be called the Fcck matrix. This more general

apcrcach to Hartree-Fcck 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 (VI-8)

an approximate one-electron model Hamiltonian of the fcrm

+
l = Eaa a = n (VI-10)
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

spin-crbital basis the nonorthoqonal atomic spin-ortitals

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 (VI-11)
k ik k ki

or the matrix fcrm


F = SES (VI-12)

where 6 is a diagonal matrix. Equation (VI-7) in ratrix

form teccaes

-1 -2
G(E) = E S S S E .

-1 -1
= (ES -6) (VI-13)









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

ISSS-F I. (VI-14)

The ccrrespondinc eigenvalue Frcblei


PC = SCE (VI-15)

can te reduced to the simpler one

F'C' = C'E (VT-16)

where

+
C' C' = C'C' = 1 (VI-17)

and


F' = -]SX (VI-18)

with X being the diagonal matrix with elements

1/2
X = (-8 ) (V1-19)
kk k

This can be considered as the diagonalization of a Fcck

matrix in a basis which is energy-weighted 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 ) (VI-20)
ki ki i k

The electron propaqator in this orthgconal tasis becomes

-1 -1
G(E) = X (EI1+XSX) IS, (VI-21)









and the elements cf the corresponding one-electron 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) (VI-22)
C kl

where C enclcses the occupied energies. An appropriate

defiriticr of t1e c-arqe and bord-crder matrix iculd in this

case be

-1I -1
P = 12ni) ) IF IE1+YSX) (VI-23)
C

This choice leads to

2
q = F = k kk i i ki

for the formal spin-orbital 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 (VI-10), one would

have tc resort to some other type of orthogonal

transformation in order to avoid a complex Fock matrix

(VT-18). 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 positive-energy

(virtual) spir-orbitals is not a straightforward task.















CHAPTER VII

THE GCPNP-CANCNTCAL HARTBEE-FOCK SCHEME








The notion of an atom in a molecule led Slater (1970) to

consider an enqrny functional obtained as an average over

multipl-ts arising from a given configuration. His

extension of this idea to configurations with fractional

cccupaticns, known as the HyFer-Hartree-Fock mathcd,

however, has been subject to some criticism because cf the

appearance of cff-diaqcral 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 off-diaqcral Lagranqiar multipliers.



A system of noninteractinq electrons described by the

Hamiltcnian


H = n (VII-1)
i i i


-45-





-46-


can Le described by the Grand Canonical partition furctior

related tc the density cFerator

S= n (1-n +2 n )/(1+z ) (VII-2)
i i ii i

%here

z = expf- (& -p)/T]. (VII-3)
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 ). (VTI-4)

In particular, the average value of the number operator is

= / 1+z ). (VII-5)
i i i

Usinq this ir expression (VII-2) gives a unique way of

defining the density operator from a given set of occupation

nurters,

p = n rl-+(2-1)n ], (VII-6)
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 electron-ccnserving (Cancrical)

average valuE of the nany-electron Ilamiltonian in the

crthonormal basis which diaqonalizes the density matrix, one

has


= h + 1/2 (J -K ) (VII-7)
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 Grand-Cancnical density operator then

, (VII-8)
ii i i

and equation (VII-3) can he written


= 1/2 3 V (VII-9)
i i i ij ij i j

where the effective one-electron energy is given by


S = h V (VTI-10)
i i i ii i

and the effective interaction energy by

V = J -K (VII-11)
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 (VII-12)
i i





-48-


2
a /aa = a& /a = v. (VIi-13)
i i i i ij

Such interpretations lead to the finite-difference

approximations tc icnization energies

(i)
d E = A (VII-1)
i i

and excitation energies

(i- -i)
A E = S + F A + V % L (VTI-15)
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 one-electron (Grand-Canonical) operator

having ttese cne-electrcn ererqies as its eigenvalues, has

matrix elements, in an arbitrary nonorthoqonal tasis, cf the

form

+
S = h + r (kllk'l')-(kl' Ik'1) ]
(VTI-16)
kl kl k'l' k' 1'

where the density matrix elements are given by

+ +
C (VII-17)
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 self-consistent

construction and solution of equation (VII-16) is what has

come to be called the Grand-Canonical Hartree-Fock (GCIF)

method, .ith the total energy given by equaticns (VII-9),

(VII-10) and (VII-11).
















CHAPTFR VITI

A SELF-CCNSISTENT CHARGE ANE
CCNFIGURATICN PRCCEDURf








Up to now we have said very little about the manner in

which the spin-crbital occupations are assigned. This is

somewhat of a sensitive subject, here and in many other

"bcildinq-block" aVproaches. This question has its roots in

the early days of valence-bond 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 ncnor-hcgcral

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 second-order 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

well-defined "valence states". Any less couplirg, however,

can lead to ill-defined 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 spin-orbitals, which frcm the beginning

puts space and spin cn similar footing. A single

configuration of spin-orbitals will, in general, not

correspond to a proper spin-coupled state, but the atomic

"valence states" remain rather well-defined, 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 perfectly-paired state to the proper atomic

states. Particularly attractive, in this regard, is the

spin-valence 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) atom-in-molecule

approach. The basic question is "hcw do we choose cur spin-

orbital occupations in our single-configuration, 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 spin-ortitals 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 electrcn-pair 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 (VIII-1)


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') (VIII-2)

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) (VIII-3)

and

(C1s) (C1s') (C2s) (C2s') (C2pz') (C2py') (VIII-U)

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 separated-atom states.

What if we krew that the molecule had a triplet2 ground

state which can not he formed by a smooth coupling of the

separated-aton 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 single-configuration

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 self-consistent 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 (VITI-1). 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

(valence-tond) 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

valence-cund structure. Such a prccedure would be very much

like the hypor-[artree-Fock 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

separated-atom 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-

so-popular automobiles dcwn here cr earth, there has teen

relatively little thecretical electronic structure %ork

reported for this first-row diatomic molecule. Restricted

Hartree-Fcck 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 cne-electron

properties with rather limited success.' Put tc our

knc!ledqe their0 has been no unres-ricted Hartree-Fcck

calculation (spin-projected or otherwise) reported on this






ISee also the natural orbital calculation of Kouba and
Chrn (1971).


-58-





-59-


odd-electron paramagneticc), open-shell system tc assess, on

a one-electror 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 Lefehvre-Bricn 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 valence-bond

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 valence-bcnd-type 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') (IX-1)

in terms cf "perfect-paired" atomic spin-orbitals 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') (IX-2)

where the 26, 46 and 2ny orbitals would be of antibonding

character and the 1i ard 1-y ortitals of rather nonbcnding

character. In our procedure, the nitrogen and oxygen

orbitals in (IX-1) 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 Hartree-Fock solutions)

ccrrespcnd roughly to the molecular orbitals in (IX-2).





-61-


The basis sets used for nitrcqer and oxygen are STO's of

double-zeta quality (Poetti and Clementi, 1974) plus d-type

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 p-type function polarized along the

bond axis (z-direction here). We chose an even-tempered 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 spin-orbital

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 valence-bond-type structure. This

is done tc eliminate any effects due directly or indirectly

to the mcd-l Hariltcnian. The result was a total energy of

-129.44 a.u., which is about 0. 1 a.u. below the estimated

Hartree-Fcck 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 0.9-c47 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
Even-terpered 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) valence-tcnd

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 charge-consistent 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 many-body-pPrturbation calculations, where





IThe terms "singlet" and "triplet" refer to 'he final
icnic states resulting in the dele+ion of a majority- or
minority-spin ortital, respectively (the "singlet" actually
being an average of singlet and +riplet states).





-65-


the Hartree-Fcck 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 Hartree-Fock 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 charge-consistent

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

half-widtl (FWHM=.r'eV) weighted by the spin-degeneracy 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 + 1-ny' 17.8 18.3

I 1-nx' 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
Lefebvre-Bricn and Moser (1966). (Adiabatic)
f
Thulstrut anc Oirn (1972). (AdiabaFtic)































Piqure 2. C-icula+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
















A-J
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

"hidden-line" picts (McTntosh, 1975) in +he yT-plane 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 spin-polarized valence-tbnd

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 spin-projected unrestricted Hartree-Fockz

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 spir-polarized X-alpha calculation of Connolly et al.
(1973) is in reasonable agreement with experiment.
'Using the same basis and Grand-Canonical Hartree-Fock.































Figure 3. "Fljddon-ljn&' Flots of the atoric crtitals
on the nitroqEn atcu. yz-r-ane (x=C.O).

















--


ii:


~liv,"









__; -: -; .1



















-*-:~ ., I" -


I I


~-~-""' '' r~ ::~T~ -.
'1



"'
i!
































Figure 4. tHilden-linel Ficts of the atomic ortitals
cn the olpqOn atcm. yz-plane 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 Hartree-Fock 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 separated-atom 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 least-squares fit tc the

Hultert-Firsctfelire (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 energy-conscious 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 surface-adscrtate

ccmplexes.i Or the theoretical side, various approaches stem

from both solid-sta'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 self-consistent 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 atom-cluster 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 nearest-neiqhtor environment, even

though these noiqhbcrs [ay not he explicitly in the cluster.

This, we feel, is a very important feature to be

incorporated in small-cluster 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 self-consistent description of an atom in the

metal. In the case of nickel, we have a face-centered-cutic

(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-


nearest-neighbor 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 nearest-neighbor

interactions cnly. The self-consistent calculation proceeds

as follows. We start with a free atom calculationi using

the hybrid sirqlE-dcuble-zeta basis given in Tatle 5 and a

spherical 4-terr model potential expression with the

exponents in able 6. We then take 13 of these atoms with

their free-atcm 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

12-free-atom environment taken fully intc account. Now

comes the first impcrtart step toward self-consistency. 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 self-consistent "atom-in-metal"

is shown in Fiqure 7, ccopared tc that of the free atom.

Cne cculd, in principle, extend this into a hand calculation






'An unrestricted Grand-Cancrical Hartree-Fock calculation
fcr the 3F state (de configuration).
zFerroiaqnetic couplirq (triplet pairing) of spins.































Fiqure f. Cluster models for the bulk, surface and
four-fold 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 double-zeta 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
Even-tempered 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 O-140eV).




-85-


NI RTOM


NI METAL


NI (100)

SURFACE





-86-


just as .ith the renormalized-atom 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 transform-deconvcluted 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 nearest-neighbor 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 self-ccnsistercy 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 four-fold 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 well-known process

(K~bylinski and Taylor, 197) It is, in fact, the

underlying process of the catalytic converters installed in

neter emissicr-ccntrolled 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

nitrcqen-cxyqen 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 X-ray (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 oxide-like 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 four-fold 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 nitrogen-oxygen 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's-rule atom). The nickel lattice constant was left

unaltered at U.7 a.u., the nitrogen-nickel 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 mod-l 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 coordinate-1 site (on

top of a nickel atom) this could have teen done with the

nine-atoa 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 yz-plare 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.)




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