Unrestricted hartree-fock molecular orbital calculations on transition-metal complexes : a detailed study on TiF 3-6

Unrestricted hartree-fock molecular orbital calculations on

transition-metal complexes : a detailed study on TiF 3-6

Citation for published version (APA):

de Laat, F. L. M. A. H. (1968). Unrestricted hartree-fock molecular orbital calculations on transition-metal complexes : a detailed study on TiF 3-6. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR162488

DOI:

10.6100/IR162488

Document status and date: Published: 01/01/1968

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-UNRESTRICTED HARTREE-FOCK

MOLECULAR

ORBITAL CALCULATIONS ON

t

-1

II

TRANSITION-METAL COMPLEXES

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UNRESTRICTED HARTREE-FOCK MOLECULAR

ORBITAL CALCULATIONS ON

TRANSITION-METAL COMPLEXES

UNRESTRICTED HARTREE-FOCK MOLECULAR

ORBITAL CALCULATIONS ON

TRANSITION-METAL COMPLEXES

A DETAILED STUDY ON TiF

~-PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HO-GESCHOOL TE EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, DR. IR. A.A.TH.M. VAN TRIER, HOOGLERAAR IN DE AFDELING DER ELEKTROTECHNIEK, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP DINSDAG

5 NOVEMBER 1968 TE 16 UUR

DOOR

FRANCISCUS LAMBERTUS MARTINUS ARNOLDUS HENRICL'S

DELAAT

GEBOREN TE NUENEN

Dit proefschrift is goedgekeurd door de promo tors Prof. Dr. G.C.A. Schuit

en Prof. Dr. Ir. P. Ros

DANKBETUIGING

Bet onderzoek, in dit proefschrift besahreven, k~am tot stand met de financi~le steun van: het Hoogewerff Fonds, de Koninklijke Shell en de Stiahting Saheikundig Onderzoek Neder-land (ZWO), waarvoor ik hier mijn oprechte dank wil uitspreken.

Bet Eindhovens Hogeschool Fonds dank ik voor de financi~le

steun, waardoor ik in staat werd gesteld deel te nemen in 1984 aan het "NATO Advanced Study Institute in Theoretical Chemistry" te Frascatie (ItaZilt) en in 1967 aan de "Summer Sahool in Theo-retical Chemistry" te Oxford (Engeland), heiden o.l.v. Prof. C.A. Coulson.

In het bijzonder gaat mijn dank uit naar het Rekencentrum van de Technisahe Hogeschool Eindhoven voor het veelvuldig ter beschikking stellen van de EL-XB computer. Zander deze hulp zou dit onderzoek in deze vorm onmogelijk zijn geweest,

Tenslotte wil ik bedanken de heer W.H.J. Kuipers voor de aorrectie van de Engelse tekst en de heer W. van Herpen voor het vervaardigen van de tekeningen.

TABLE OF CONTENTS

INTRODUCTION • • • • 1.1 Historical review

1.2 Outline of the present work References

2 HARTREE-FOCK METHODS

. . .

.

2.1 Conventional Hartree-Fock scheme. 2. 1 • 1 General theory • • • • • • • 2.1.2 The Hartree-Fock equations • • 2.2 Unrestricted Hartree~Fock scheme

2. 2. 1 General theory • • • • • • 2.2.2 The LCAO-MO approximation

2.3 Projected-unrestricted Hartree-Fock scheme 2.4 Symmetry orbitals • • • •

2.5 Slater-type orbitals • References • • • • • •

3 SINGLE ANNIHILATION FOR A SINGLE DETERMINANT WAVE-FUNCTION • • • • • • • • • • • • • • •

;:

3.1 Average expectation value of the S -operator

• " • 9 • 9 12 14 17 17 17 20 22 22 24 25 26 30 32

3.1.1 <S2> before single annihilation • • • • 34 34 34 36

37

3.1.2 <82> after single annihilation • • •

3.2 Total electronic energy • • • • • • • • • 3.3 Charge-density and spin-density functions

References • •

'· 37

39

4 MOLECULAR INTEGRALS WITH SLATER-TYPE FUNCTIONS • • 40 4 • 1 General concepts • • • • • • • • • • 40 4.1 .1 The spheroidal coordinates • • 41 4.1.2 The V _{n mp 1} (t,·r)-functions. • • • 43 4.1. 3 The Gaunt coefficients • • 45 4. 2 One-centre integrals • • • • • 49 4. 3 Two-centre integrals • • • • • 52 4.3.1 Two-centre one-electron integrals 52 4.3.2 Two-centre two-electron integrals 55 4.4 Three-centre one-electron integrals • • • 61 4.5 Approximation of three-centre and four-centre

4.6 Relation between integrals with real and

complex functions • • • • • • • • • • • • • • • • • _{67 .} 4.7 Description of a real orbital in a rotated coordinate

system • • • References • •

68

• • • • • 70

5 A STUDY OF THE APPROXIMATIONS IN AN UNRESTRICTED HARTREE-FOCK CALCULATION ON TiF~- • • • 5,1 Survey of the computation scheme • •

5.2 Selection of basis sets for Ti and F • • • • • 5,3 The treatment of the core-electrons

5.4 Influence of the three-centre and four-centre two-electron integrals on both the total electronic energy and the 1 ODq-parameter • • • •

72 72 76

eo

84 References ·. • • • • • • • · • • • • • • • • 89

6 SOME COMPUTED QUANTITIES OF TiF~- FOR VARIOUS BASIS SETS

7

8

AND AT VARIOUS METAL-LIGAND DISTANCES· • • • • • 90 6.1 General introduction • • ~ • • • • • • • 90 6.2 computed quantities with the unprojected single

determinant wave-function • • • • • · • • 99 6.2.1 Total electronic energy of the ground state

2_T

2 and first excited state

2_{E • • • • 99}

g g

6.2.2 Crystal-field splitting parameter (10Dq) • 104 6. 2. 3 Orbital populations · • • • • • • • • • • • 1 07 6.2.4 Testing the Wolfsberg-Helmholz approximation • 109 6.3 Computed quantities before and after spin projection 109

. 2

6.3.1 Average expectation value of the S -operator 109 6.3.2 Charge-density and spin-density functions • • • 111

References • • · • • • · • • • • • • • • • · · •

DISCUSSION References CONCLUSIONS

LIST OF PRINCIPAL SYMBOLS SUMMARY SAMENVATTING LEVENSBESCHRIJVING • 115 116 121 123 127 131 133 135

1 INTRODUCTION

In this chapter a historical review is given of the

calcu-lations carried out in the last decades in order to obtain

nu-merical data for some properties of the first-row

transition-metal complexes. This survey does not pretend to be complete,

but mainly refers to an arbitrary number of non-empirical and

semi-empirical calculations representative for the progress in

the theoretical analysis of the electronic structure of

com-plexes.

Next, the limitations in these treatments are quoted and the

main features of the investigation in this thesis are discussed.

1.1 HISTORICAL REVIEW

Most of the interesting features of the physics and chemis-try of the first-row transition-metal ion complexes are related to the ap~itting of the energy levels of the 3d-orbitals of a transition-metal ion (central ion) under the influence of a

"arysta~-fie~d" caused by the ligands. In oatahedraZ complexes the orbitals appear to split into two sets: one set of 3d-orbitals (dx2-y2' dz2) pointing toward the ligands, the other set of 3d-orbitals (d , d , d ) being located between the

xy xz yz

ligands. An elegant description of this phenomenon can be given by using group theory. The two sets of 3d-orbitals classified within the group theory according to the irreducible representa-tions e and t₂ are further denoted by e and t₂ -orbitals.

g g g g

In the special case of a system with a single 3d-electron, the energy of the system with the electron in a e -orbital will _g be above that with the electron in a t₂g-orbital. The energy difference is called the "arystaZ-fie~d spLitting" parameter 11

or 10Dq.1 _{This quantity can be evaluated experimentally f·rom the}

position of the crystal-field absorption band in the electronic spectrum of the complex.

The first attempts to evaluate the parameter 10Dq without using empirical data were made as early as 1939 by Van Vleck2 and Polder.3 _{The case chosen by Van Vleck was a central cr 3+ion}

surrounded octahedrally by six water molecules1 Polder studied a tetragonal surrounded cr2+ion. They used a point-ahaPge or point-dipoZe approximation for the (negative) ligands and their calculations gave the correct sign for 10Dq. To some extent this result is obvious. However, when Kleiner4 extended the com-putations by using a deZooaZised model (Slater-orbitals5 ) for the ligands instead of point-charges, he obtained the wrong sign for 10Dq, The reason was that in his model the positive nuclear charge of the ligand attracted the e -electrons more than the _g ligand-electrons repelled them.

tained by Tanabe and Sugano6 mechanical exchange integrals.

The correct sign was again ob-who also included the

quantum-Sugano and Shulman' made a detailed computation for the oc-tahedral NiF:- complex at a metal-ligand distance R = 4.014

R

which is equal to the Ni-F distance in the K

3NiF3 crystal. Like

all authors mentioned before, they started from an ionia model for the complex but allowed the metal~functions to mix with li-gand functions and permitted different wave functions (composed of Slater-type orbitals) for the spin up (a) and spin down (6)

electrons (spin-unPestrioted method8 ). The Ni2+ electrons that have been taken into account are the 3d-electrons only. In spite of the great number of simplifications and approximations ap-plied in their calculation, they obtained a crystal-field split-ting parameter 10Dq

=

6,350 cm-1 which is in reasonably good agreement with the experimental value (7,250 cm-1).

But as has been pointed out by Watson and Freeman9and

Sim~-'

nek and SroubeklO their method appears to be theoretically in-correct, since in accordance to the arguments of Watson and Freeman the ("unpaired") bonding orbitals must be considered in-stead of the (unpaired) anti-bonding orbitals to characterise

4-the covalency in 4-the NiF₆ complex. The value of 10Dq according

-1

to Watson and Freeman's method was only 2,815 em and 2,760 cm-1 dependent on the choice of the method of approximation.

An attempt by Sugano and Tanabe1_{l to elucidate the}

discrep-ancy between the result of Sugano7 and that of Watson9 failed. O:ffenhartz1 2 calculated the 10Dq- parameter for NiF~- using the ionic Hamiltonian model and obtained a value of 4,040 cm-1•

In contrast to this set of calculations for the NiF:- com-plex, all carried out with Slater-type orbitals (multi-centre basis set), Ellisl 3 and Ros1_{~ used a one-centre basis set, 1 5}

i.e. all wave functions of the complex were described in the same coordinate system. They performed a self-consistent-field

4-(SCF) calculation on NiF₆ and included nearly all electrons of the complex. They obtained a value of 10,800 cm-1 for 10Dq, which is above the experimental value.

Richardsonl6 recently made a set of (spin-restricted) cal-culations for the octahedral first-row transition-metal fluo-rides and described these systems with Slater functions for the fluorine and Slater-type functions17 ' 18 for the central-metal ion. Also in this work approximations have been applied to de-crease the computation time. His calculations for TiF~- yield values for 10Dq which were quite near to the experimental value, dependent on the choice of the basis set. Some more or less sat-isfactory results were obtained for the other fluorides.

Fenske et al.l9 carried out a calculation for the transi-tion-metal hexafluorides and adapted some parameters to obtain the correct 10Dq. Analogously, the octahedral and tetrahedral transition-metal chlorides 2 0 were also considered. However, the method by which they determine the value of 10Dq is questionable as will be shown later.

At the same time others tried to evaluate the crystal-field splitting parameter by means o~ semi-empiriaaZ methods (see, for example, ref. 21,22), i.e. certain integrals for the potential and kinetic energy of the ions in question are approximated by 'the vaZenae state ionisation energies or potentiaZs (VSIE or VSIP) of the separated ions. These quantities can be obtained from the tables of Moore. 23 Moreover, one uses in the semi-em-pirical method the Wolfsberg-Helmholz2_{~ or Ballhausen-Gray 25}

approximation.

Out of this set of computations we shall quote: a calculation on

2+ 25 3- 2-

3-.VO(H20)5 by Ballhausen and Gray, _{on TiF 6 , VF6 and VF6} by Bedon et al., 21 ' 26 on chloroammine complexes of co 3+ by Yeranos and Hasman, 27 on NiCl!- by Valenti and Dahl,2a on Fe(CN):- and Co(CN)~- by Alexander and Gray, 29 on TiF~- by Fenske30 and a

general paper by Basch et al.22 _{for the octahedral and}

tetrahe-dral transition-metal complexes.

Other semi-empirical calculations using some non-empirical quantities instead of VSIE's or VSIP's are: on TiCl~- by Oleari

39 2+ )2+ 32

et al., on Cu(NH

3)6 . and Cu(H 2

o

6 by Roos and on some chlo-rides of Cu by Ros and Schuit.33 ·

Ciullo et al.34 _{have worked out a general method for the}

transition-metal complexes which uses empirical parameters as well as exact values for the representation of the repulsion between two electrons, althoug~ up.to now no application for any special case has been reported.

1.2 OUTLINE OF THE PRESENT WORK

As appears from section 1.1 there has been no calculation on the first-row transition-metal complexes giving us a good in-sight into the reliability of the various approximations ap-plied. Moreover, a lot of authors either evaluate only the crys-tal-field splitting parameter 10Dq or use the experimental value of 10Dq as a basis for determining some auxiliary parameters in their calculations.

A correct description of the electronic structure of tran-sition-metal complexes (crystals) becomes more and more desira-ble in order to obtain a better insight into various properties and quantities such as: the stability of a complex (crystal), the crystal-field splitting parameter 10Dq, the charge transfer bands, the total energy as a function of the metal-ligand dis-tance,the ionisation energy, the charge-density and spin-density functions, the hyperfine interaction, direct and super exchange interactions.

Hence this thesis describes a general method yielding re-sults for the total energy, the 10Dq, the orbital populations, the average expectation value of the

s

2-operator, the total

en-ergy as a function of the internuclear distance, and the charge-density and spin-charge-density functions of a complex. This general method will henceforth be indicated by "spin and symmetr>y-un-restricted Hartree-Fock (UHF) method", in which a linear

combi-nation of Slater-type atomic orbitals (LCAO) will be used to construct the molecular orbitals (MO) .·• To obtain in this method a self-consistent-field solution an iteration procedure must be followed, which was not done in most of the non-empirical meth-ods7'9'11•12 quoted in section 1.1.

The spin and symmetry-unrestricted Hartree-Fock scheme will

I

3-be applied on ad -system: the octahedral TiF₆ complex. In this calculation a single determinant wave function will be used. The influence on the results of the following points will be discussed in relation to this more or less arbitrarily chosen complex:

(1) approximation or neglection of various integrals occurring in the computation method~

(2) the iteration proceduret

(3} choice of the basis set implying the number of basis func-tions together with their orbital exponentst

(4} the electronic configuration of the complex~ (5) the metal-ligand distance.

The UHF-method yields a wave function which is generally not an eigenfunction of the

s

2-operator (no pure spin state), However, a simplified spin-projection technique (single annihi-lation) discussed by Amos et al.3 5•36 and Sutcliffe37 _{will, in}

their opinion, correct the wave function into a relatively good eigenfunction. They used this projection method for aromatic radicals and the NH₂ free radical respectively.

To get an idea about the effect of the single annihilation on the single determinant wave function of a transition-metal com-plex, some properties (the average expectation value of

s

2, the charge-density and spin-density functions) will be evaluated be-fore and after spin projection.

The complete method implying the definitions of all terms used before has been worked out in chapters 2 and 3, the latter mainly dealing with the spin-projection method.

All types of integrals occurring in the calculation are

ed in chapter 4, where the accessory expressions will be derived as well.

Chapter 5 deals with the computation scheme and a detailed cal-culation on the octahedral TiF~- complex. Moreover, the effect of approximation or neglection of integrals on the following re-sults will be considered: the crystal-field splitting parameter 10Dq and the total energy of a complex. In chapter 6 the influ-ence of the iteration procedure, the basis set and of the metal-ligand distance is dealt with.

The discussion of all results obtained, the comparison with ex-isting calculations as well as ~ number of conclusions and pos-sible extensions of the computation method developed can be found in chapters 7 and 8.

Concerning the experimental data in the literature for TiF~- we refer to the work by Siegel38 _{in which the Ti-F}

dis-tance in TiF

3 was found to be 1,97

i.

Bedon et al.21 observed the absorption spectra of NaK

2TiF6, Na2KTiF6 and (NH4)3TiF6

sol-ids in KCl and KBr pellets and found values of 10Dq for a hypo-thetical octahedral symmetry of 17,500 cm-1 , 17,450 cm-1 and 17,060 cm- 1 respectively.

The ab initio calculations on the the Electrologica XS

spin and symmetry-unrestricted Hartree-Fock octahedral

TiF~-

complex were performed on (EL-X8) computer (high-speed memory: 32,000 wordSJ drum: 524,000 words) of the Computer Centre of the Tech-nological University at Eindhoven.

The complete set of computer programmes for the evaluation of the great number of integrals (about 85,000 for each calcula-tion) as well as the SCF-computer programme have been written in ALGOL 60.~ These computer programmes4 0 as well as the integral values41 _{used in the investigation are not included in this}

thesis.

REFERENCES

l, Dunn, T.M,, McClure, D.S., and Pearson, R,G., Some Aspects of Crystal Field Theory, Harper and Row, New York, Evanston and London, and John Weatherhill, Inc., Tokyo (1965), ch.l.

2. Van Vleck, J.H., J.Chem.Phys. 7, 72 (1939). 3. Polder, D., Physica 9, 709 (1942).

4. Kleiner, W.H., J.Chem.Phys. 20, 1784 (1952).

5. Slater, J.C., Phys.Rev. 36,

57

(1930).

6. Tanabe. Y., and Sugano,

S:,

J.Phys,Soc. Japan

!!•

864

(1956).

7. Sugano. S., and Shulman, R.G., Phys,Rev. 130. 517 (1963), 8. Pople, J.A., and Nesbet, R.K •• J.Chem,Phys:-22, 571 L

(1954).

-9. Watson, R.E., and Freeman, A.J., Phys.Rev. 134, 1526

(1964).

-10. Simanek, E •• and Sroubek,

z.,

Phys.Status Solidi 4. 251

(I 964) •

-11. Sugano,

s.,

and Tanabe, Y., J.Phys,Soc. Japan!£, 1155

(1965).

12. Offenhartz, P.O., J.Chem.Phys. 47, 2951 (1967). 13, Ellis, D.E •• MIT Ph.D.Thesis (1966).

14. Ros. P •• (private communication).

15, Ellis. D., and Ros, P., MIT Quarterly Progress Report, Solid-State and Molecular Theory Group 58, 42 (1965); 59, 51 (1966).

16. Richardson, J.W., (private communication).

17. Richardson, J.W., Nieuwpoort. W.C., Powell, R.R •• and Edgell. W.F., J,Chem.Phys. 36• 1057 (1962).

18, Richardson, J.W. Powell, R.R., and Nieuwpoort,

w.c .•

J.Chem,Phys. 38, 796 (1963),

19. Fenske, R.F., Caulton, K.G., Radtke, D.D., and Sweeney,

c.c.,

Inorg.Chem, 5, 951 (1966); 5, 960 (1966).

20. Fenske, R.F., and-Radtke, D.D., Inorg.Chem. 7, 479 (1968). 21. Bedon, H.D •• Horner, S.M., and Tyree Jr •• S.Y., Inorg,Chem.

3. 647 (1964).

22. Basch~ H.,Viste, A., and Gray. H.B., J.Chem.Phys. 44, 10

(1966).

-23. Moore, C.E., Atomic Energy Levels, Circular of the Nat.Bur, Std, No. 467 (1949), Vols. I,II, and III.

24. Wolfsberg, M,, and Helmholz, L., J,Chem.Phys. 20, 837 (1952).

25. Ballhausen, C.J., and Gray, H.B., Inorg.Chem.

J:

I l l (1962).

26. Bedon, H.D., Hatfield, W.E., Horner, S.M., and-Tyree Jr., S,Y., Inorg.Chem. 4, 743 (1965),

27. Yeranos, W.A., and Hasman, D.A., Z.Naturforschg, 22a. 170

(1967).

-28, Valenti, V., and Dahl, J.P., Acta Chem.Scand. 20, 2387

(1966).

-29. Alexander, J,J., and Gray, H.B., Coordin.Chem.Rev. ~. 29

(1967).

30. Fenske, R.F., Inorg.Chem. ~. 33 (1965).

31. Oleari, L •• Tondello, E •• Di Sipio, L., and De Michelis, G.,

Coordin.Chem.Rev. 2, 45 (1967). ·

32. Roos, B., Acta Chem.Scand, 20, 1673 (1966).

33, Ros, P., and Schuit, G,C.A.~Theoret.chim. Acta (Berl.) ~. 1

(1966) •

34. Ciullo, G., Furlani,

c.,

and Sgamellotti. A., Coordin.Chem.

Rev. 2 • I 5 (l 9 6 7) •

35. Amos, A.T.~ and Hall. G,G .• Proc,Roy.Soc. (London) 263A, 483

(1961).

37. Sutcliff*• B.T., J,Chem.Phys. 39, 3322 (1963).

38, Siegel,

s.,

Acta Cryst, 9, 684~1956).

39. Backus, J,W,, et al., Numerische Mathematik 4, 420 (1963). 40. DeLaat, F,L.M.A.H., Complete Set of Computer Programmes for

Unrestricted Rartree-Fock Calculations (ALGOL 60), Technische Hogeschool Eindhoven (1968), (unpublished), 41. DeLaat, F.L.M,A,H., Integral Values of Unrestricted

Rartree-Fock Calculations on TiFg-, Technische Hogeschool Eind-hoven (1968}, (unpublished}.

2 HARTREE-FOCK METHODS

This chapter starts with a brief review of some basic

con-cepts followed by the conventional Hartree-Fock method and the

accessory Hartree-Fock equations. Some restrictions in this

method are rejected and the then resulting unrestricted

Hartree-Fock method discussed. Next, the Hartree-Fock equations are

transformed into a pseudo-eigenvalue problem by choosing an

ex-pansion for the space part of the one-electron functions. This

general case will lead to the method of linear combination of

atomic orbitals, which is used in the molecular orbital method.

An iteration scheme for obtaining a self-consistent solution for the eigenvectors is elucidated.

In connection with the evaluation of spin properties, the

projected unrestricted Hartree-Fock method is quoted.

A possibility to reduce the pseudo-eigenvalue problem with the

help of the symmetry-adapted orbitals will be briefly outlined.

~n the last paragraph we consider the Slater-type function which

has been chosen for the description of the one-electron wave

function,

2.1 CONVENTIONAL HARTREE-FOCK SCHEME

2.1.1 General theory

In this paragraph a brief survey will be given of the basic concepts of the Hartree-Fock (HF) method; a complete treatment can be found in the papers of· refs. 1-6.

For a many-electron system (atom, molecule or crystal) we have the Schr~dinger equation H~ = E~, where H is the Hamilto~ nian operator and ~ the wave function or many-electron wave function of the system. If ~ is an eigenfunction of H,the eigen-value E represents the energy of the system.

Generally, the Hamiltonian operator contains all kinds of elec-trostatic and magnetic interaction terms such as a term repre-senting the repulsion of the nuclei, the influence of an exter-nal field, the spin-orbit coupling.

The Hamiltonian ope~ato~ H with only the electrostatic

of the form:t

H (1 I 2 I • • • ,N) (2 .1)

where k1l7 a,B,y: electron and nuclear indices respectively;

Ak Laplace operator;

Z _a,!l,y the nuclear charge of nucleus

a,a,

and

y

respectively;

the distance between the electron k and nu-cleus Yl

the distance between the electrons k and 1, and the distance between nuclei a and !l re-spectively.

The operators in the Hamiltonian H can be divided into and two-electron operators h(k) and h(kl) respectively. The

one-:...1

electron operators are -~Ak and rky with the physical interpre-tation of kinetic energy of electron k, and the interaction of electron k with nucleus y respectively. A two-electron operator

-I

is rkl representing the interaction of the electrons k and 1. The Hartree-Fock approximate wave function ~ for a N-elec-tron system is assumed to be a (normalised) anti-symmetrised produat of N (orthonormal) one-electron wave functions, which can be denoted in a determinantal form, i.e. the single Slater determinant: '!'(1,2, ••• ,N) 1(11 (1) I/J2(1) ••• t1JN(1)

w

₁ (2) 1/> 2 (2) ••• 1/JN(2) ( 2. 2)

In the one-electron wave function

wi

(k) i refers to the function indices, k to the particle indices.

~he Pauli principle requires:

(1) two electrons cannot occupy identical orbitals1

(2) the wave function should be anti-symmetric in the electrons. The function of equation (2.2) satisfies both conditions.

The totaL energy of a many-electron system which is repre-sented by the total wave function ~ is given by:

(2.3)

since ~ (eq. 2.2) is a normaLised function, i.e.

H* ~ d-r

=

<~I~> = (2.4)

The integration T is taken over all space and spin coordinates

of the N-electrons, and for the integrals the bracket-notation of Dirac7 _{is used. The asterisk indicates the complex conjugate}

of the total wave function ~.

The assumption that the set {~i} is orthonormal

<w.l~.> = o(i,j)

l. J

(2. 5)

with o(i,j) the Kronecker delta function, is no contraint for the general solution, since the tjli's are linearly independent and there consequently always is an orthogonal transformation8

allowing the transformed set of wave functions to form an ortho-normal set. This transformation does not affect the expectation values of the total wave function~ (see ref. 9).

Now, we can reduce the expression for the total energy of a

N-electron system by substituting equation (2.2) in (2.3), using the orthonormality of set {ljli} (eq. 2.5):

N E L. <w.<t>lh<I>Iw.<I>> l. l. l. N + ~

L

<1jl.(l)ljl.(2)jh(I2)(1-Pl2)1tjl.(l)ljl.(2)> i,j l. J l. J (2.6)

2.1.2 The Hartree-Fock equations

In the Hartree-Fock method, the total energy E is to be made stationary with respect to any infinitesimal variation of the one-electron function ~i(k) subject to the orthonormality of set {Jjl.}. This va'l'iational p't'inaiple yields after some algebraic

l.

manipulations the following Ha'l't'l'ee-Foak equations (for details, see ref. 9,1 0) :

(2.7) The eigenvalues ~

₁

•s being the Hartree-Fock one-electron ener-gies.

The Hartree-Fock operator (or Hartree-Fock Hamiltonian) F in these equations is given by (cf. ref. 9):

F (1) G(1) J. (1) J K. (1} J h(1) + G(1) ~ J.<1>- ~ x.C1) J J J J <lji. (2) ih(12) !~. (2)> J J <ljoj (2) ihC12JP 12J!joj (2)>

where h (1): one-electron operator;

G (1): total electronic interaction operator;

Jj(1): Coulomb operator; K. (1): exchange operator. J (2.8a) (2.Bb) (2.8c} (2.8d)

The general procedure for solving the HF equations is iterative and called the Hartree-Fock self-aonsistent;...field (SCF) method. This subject will be discussed in detail in section 2.2.

The one-electron function ljoi{k) can be written as a product of a space part (o'l'bital) and a spin part since the total Hamil-tonian and consequently the HF HamilHamil-tonian (in our case) does not contain spin terms:

n₁(s) is an eigenfunction eigenvalue of szi: ± ~.

into equations (2.8a-d)

(2.9) of the spin operator szi with possible After substitution of equation (2.9)

and application of the orthonormality properties of the spin functions ni(s), the energy expression

{2.6) becomes for a system'with closed-shell structure (see for the definition Roothaan9):

EHF

₌

_{2}: H. +}

_1:

_{(2J .. -K .. )}

i ~ i. j ~J ~J

=

1:

(Hi+E:i} with i,j

=

1, ••• ,N/2 (2 .1 0 ) i where H. = <<j>.(1llh (1ll<f>.(1}> (2.10a) ~ ~ ~ Jij

=

«~>· ~ <1>

!J.

J <1> I<~>· ~ <1l> (2.10b) K ..

₌

<<j>. (1)

!K.

(1) I<~>· (1)> (2.10c) ~J ~ J ~

For open-shell systems a similar energy expression has been worked out by Roothaan.1 0

The conventional HF method described in this paragraph has some restrictions:

{ 1 ) the method is spin-~estriated, _{i.e. for the set}

*

_{<f>i} 01. _and

{<f>~} holds <4>~14>~>

=

o{i,j). For the spin-unrest~icted

~ ~ J '

method the orbitals 4>~ may be different from 4>~1

~ ~

(2) the method is symmet~y-restricted (it consists of symmetry-restricted orbitals), i.e. the orbital 4>i is a basis func-tion of an irreducible representafunc-tion of the symmetry group of the system.

For symmetry-unrestricted orbitals, 4>i can be any function satisfying the one-electron HF equation for that system7

{3) both the restricted (conventional) and the unrestricted HF method still contains correlation erro~s (cf. 11,12). How-ever, the properties which will be investigated in this thesis are hardly sensitive for these errors.

As a consequence of the constraints (1) and (2), the restricted HF method will give a poor treatment of systems with unpaired electrons. A disadvantage of the unrestricted HF method is the fact that the total wave function will generally be no eigen-function of the total spin-operator

s

2• However, this disadvan-•4>~. 4>~ indicate the spatial part of a one-electron wave

func-~ ~

tage can be eliminated for the greater part by a spin projection technique. This subject will be studied in section 2.3.

2.2 UNRESTRICTED HARTREE-FOCK SCHEME

2.2.1 General theory

The spin-unrestricted Hartree-Fock method2 '3' 4 '13 allows

different orbitals for electrons with different spins and there exists a one-to-one correspondence between the electrons and the

one-electron wave functions. Where we speak of the unrestricted

Hartree-Fock (UHF) method we' mean the spin- and symmetry

un-restricted Hartree-Fock method.

The N-electron wave function in the form of a single

deter-minant with p orbitals 4~ occupied by electrons with a-spin, and

l.

q orbitals ¢~ occupied by electrons with S-spin is for the UHF

l.

method (denoted in a brief notation) :

'fUHF(1,2, ••• ,N)

=

(Nl)-j •

• d et { ¢ ~ ( 1 ) a ( 1 ) , •••

,<I>; (

p) a ( p) ~ .P ~ ( p+ 1 ) S ( p+ 1 ) , ••• , <1>! ( N) S ( N) } ( 2 • 11 )

The abbreviation "det" stands for the Slater-determinant form

(cf.eq.2,2).

The total energy is in this general method, suitable for open

and closed shell systems, analogous to equation (2.10):

a+S a+S a S

a S a+S

L

H. + ~

L

J .. - ~(

L

+

L

)K .•

i l. i,j l.J i , j i,j l.J

(2.12)"'

L

1

L

and

L

indicate summation over a-, o-, and all occupied

or-bitals respectively. For the meaning of the integrals H~, J ..

L l.J

and K .. see equation (2.10a-c). Conform to paragraph 2.1.2 the

l.J

Hartree-Fock equations can be deduced for the set {¢~} as well

l.

as for the set {¢~}.

1

Solution of the Hartree-Fock equations for complexes is

*

The differences between the factors occurring in equation (2.

10) and those in (2.12) are caused by the fact that the

summa-tion is in the first instant taken over the orbitals (N/2) and in the second over the electrons (N).

such a difficult mathematical problem that i t is still out of

the question. An approximation often used in this case, is the

expansion method,2 _{in which an orbital ~i can always be expanded}

in terms of some complete set of basis functions xt which are

assumed to be normalised. This approach is written as follows:

~~

_I

Ct Ct xtcti x.c. ~ t - -~ ~~ fl _x.c~ (2 013) ~ ~ xtcti

-

-~ t

in which x = (x₁, ••• ,xn) a row vector and £i = (Cli'''''Cni) a

column vector.

By substituting these equations in the and following the method of Roothaan for

Hartree-Fock equations

each set {~?l and ~~~},

~ ~

i t is easy to

pr>obZems:

show that this gives rise to the pseudo-eige>Z?''llu<J

(2 .14)

where the matrices Fa= H +Get= H + J - Ka, pfl

=

H + Gfl H + J

- Kfl and S (overlap matrix) defined by their elements are:

s

<x (1) lx (1)> (2 .14a) rs r r H <x (1) I h(1) lx (1) > (2 .14b) rs r s J

I

( p +Q ) < X ( 1 ) X ( 2 ) I Y' ~ ~ I X ( 1 ) X ( 2 ) ' (2 .14c) rs t,u tu tu r u s t KCI

_L

_{Ptu<xr(1)xu(2)}

_lr>~~lxt(1)x

₈

_(2)>

_{(2. 14d)} rs _t,u Kfl rs

~

Qtu<xr(1)xu(2)

lr>~~lxt(1)xs(2)

t,u (2. 14e)

P and Q, density matrices for the electrons with ~- and B-spin

respectively, are defined by:

( 2. 15)

The pseudo-eigenvalue problem can be solved by the method

described by Wilkinson14 ' 1 5'l6 and a complete computer

The total electronic energy becomes with the density matri-ces P and Q: tr(PH} + tr(QH) + ~tr(PGa} + ~tr(QGB) a S ~ {tr(PH) + tr(QH) +

L

e~ +

L

e~} i 1 i 1

where tr is the trace of the matrix.

2.2.2 The LCAO-MO approximation

(2.16 ) (2.16a)

The choice of a complete basis set in the expansion method

(eq. 2.13) will in many cases be impossible from a practical

point of view (computer capacity and computer time). One usually

takes the set of atomic orbitals of the separated atoms or ions

of the pertinent system. This approach of linear aombination of

atomia orbitals (LCAO) has been described in the paper of

Root-haan9 who used i t in the moleauZar orbital (MO) method.

The matrices Fa and FB are dependent on the density

matri-ces P and Q as can be seen from equation (2.14a-e), which in

their turn depend on the column vectors c~ and c~. To obtain a

-1 -1

self-consistent solution for the P and Q matrix, we have to use

an iteration procedure based on the following steps:

(1) assume a set of coefficients c~, c~, satisfying the

neces-- 1 - 1

sary orthonormality conditions (eq. 2.5}, and compute P and

Q:

(2) calculate the matrices J, Ka, KB according to equation

(2.14a-e) and then Fa and FB:

(3} solve simultaneously the equations (2.14} for the electrons

with a- and S-spin:l7

determine from the eigenvalues a new set of eigenvectors c~

- 1 (4}

and c~,1_-1 7 and recompute P and Q:

(5} repeat step 2 to step 4 until the computed and assumed

den-sity-matrices P and Q agree within a certain limit.

A possible ariterion for self-consistency is:

UHF

for two iteration-cycles in succession must be below a

. 1 -5 t

given lim1t, e.g. 0 a.u.;

(2) the change in each element of P and Q for two iteration

cy-cles in succession must be below a given limit, e.g. 10-4 •

2.3 PROJECTED-UNRESTRICTED HARTREE-FOCK SCHEME

The single Slater determinant wave function in the

un-restricted Hartree-Fock method does not usually represent a pure

spin state, i.e. i t is no eigenfunction of the total

spin-opera-tor

s

2• Here, we can represent the total wave function by a

linear combination of pure spin states:

"'UHF

=

~

C 'I'

'

L

s '+m s '+m

m=O

(2.17)

If we assume p > q the lowest spin component s

=

s'

=

~(p-q)

and the highest value of the spin s = s' + q. It has been shown

by Amos and Hall13 that the coefficients C _s '+ _m decreases rapidly

at an increasing value of m. For obtaining a pure spin state

with s

=

s•, the components with a spins> s' must be removed.

To construct this spin eigenfunction of muLtiptiaity (2s'+1) the

following spin projeation operator18 _{can be applied:}

rr

{s

2

-k(k+1) }

kfs' s

1

_{s

1_+1)-k{k+1) (2 .18)

The average expectation value of the total spin-operator

s

2

de-noted by <S2>sd becomes after spin projection:

2

<S >PUHF

f'l''*s2'1'' dT

f'¥'*'¥ dT

(2 .19 )

in which'¥' = os, '¥UHF. (2.19a)

The reduction in equation (2.19) is a consequence of the fact

that Os' commutes with

s

2; moreover, Os, is idempotent,

~.e.

tThroughout this thesis the atomic units (a.u.) will be used for

energy and length: I a.u. energy= 27.2107 eV

2

0

8,

=

08, . However, this spin-projection technique is so far not

tractable for a larqe system and we shall therefore only inves-tiqate the effect of sinqle annihilationl9t 20 •21 _{of the}

compo-nent with spin multiplicity (2s'+3).The single annihilator As'+J becomes then:

s

2-(s'+l)(s'+2)

As'+l

=

-2(s1_+l) (2.20)

and will remove the component with spin s

=

s' + 1 from the to-tal wave function wUHF.

The expressions for several quantities before and after sinqle annihilation are qiven in chapter 3.

2.4 SYMMETRY ORBITALS

The solution of the pseudo-eiqenvalue problem (eq. 2.14) may entail difficulties for larqe matrices Fa and F 6 with re-spect to the necessary computation time and the capacity of the hiqh-speed memory of the computer. Moreover, the computation time of the iteration process will increase as a consequence of the qrowinq number of integrals (eqs. 2.14a-e) which must be multiplied in each iteration-cycle by the elements of the densi-ty matrix.

Now, we will try to reduce the Hartree-Fock matrix F into a num-ber of independent "blocks". This subject can be investigated with the help of group theory.22-2s

The system (complex, molecule) has a certain symmetry which can be indicated by its accessory point group. From the set of basis functions

xt

we can construct linear combinations that transform according to the irreducible representations of the point group of the system in question. This classification is performed with projection operators.25 The linear combinations are called "symmetry orbitaltJ", together they form a symmetry-adapted basis set.

Use of this set of wave functions will split the original matrix Fa and FS into a number of smaller independent matrices(blocks). The size of each matrix is determined by the number of new basis functions which has be.en classified in the relatinq irreducible representation.

We will illustrate this concept for a pure octahedral com-plex

MX:-

(symmetry group Oh) with the atomic orbitals: d, s, p as basis functions for the central metal-ion M and likewise 2s, 2p for the ligands X. In this case the size of the Fa and FB-matrix will be 33 x 33. The classification of the basis func-tions according to the irreducible representafunc-tions of Oh can be seen in table 2.1.

Table 2.1 Symmetry orbitals for a complex with symmetry group Oh

•

irreducible representation e g metal orbitals 6 d 2 2 X -y d 2 _z d _xz d _yz d _xy combinations of ligand orbitals• 61 + s2 + 63 + 64 + 65 + s6 -xt - Y2 - z3 + x4 + Y5 + z6 /3( _{s 1 - 62} + 6₄ - 6₅ ) /3(-xt + Y2 + x4 - Y5 ) -s 1 - s 2 +263 - 64 - 65 +2s6 xi + Y2 -2z3 - x4 - Y5 +2z6 -zl + x3 + z4 - x6 z2 - YJ - z5 + Y6 - z3 - z6 zl + z2 + z4 + z5 Yt + x2 - Y4 - xs Yt - Y3 + Y4 - y6 - x2

+

x3 - xs + x6 -zl

+

z2 - z4

+

z5 si,xi,yi and zi stand for 2s,2px,2py and 2pz respectively on centre i.

Here, the coordinate axes on the different centres of the ions have all been chosen parallel to the main axis on the central metal-ion (see fig. 2.1).

z

Ys

Fig. 2.1 The choice of the axes for an octahedral

complex (metal on position 0, ligands on positions 1-6)

In closed-shell systems and open-shell systems treated by the method of Roothaan,9'10 _{the Hartree-Fock operator F(1) has}

the same symmetry as the total Hamiltonian H of that system. In the spin- and symmetry-unrestricted HF method this needs not necessarily be the case, however. For example, the ground state of a d1-system (octahedral complex with oh-symmetry) has 2T₂₈ -symmetry, whereas the Hartree-Fock operator of a component of 2_T

28 will have 2_B

28-symmetry (D4h) in the UHF method.

The classification of the basis functions according to the irre-ducible representations of o₄h is shown in table 2.2.

The time required to solve the eigenvalue problems of all the small matrices is less than that for the 33 x 33 matrix. However, the use of symmetry orbitals in the computer programme is a restriction of the general character of the programme and

Table 2.2 Symmetry orbitals for a complex with symmetry group D₄h

irreducible metal combinations of

representation orbitals ligand orbitals

alg dz2' s sl

+

52

+

54 + 55 -x, - Yz + x4 + Ys 53 + 56 - z3 + z6 big d 2 X -y 2 sl - 52 + 54 - 55 -x, + Yz + x4 - Ys a2g Yt - x2 - y4 + xs a2u Pz 53 - 56 - z3 - z6 z, + z2

+

z4 + zs b2g d _xy Yt + x2 - y4 - x5 b2u zl - z2 + z4 - z5 e _g d _{xz zl - z4} x3 - x6 d yz z2 - zs y3 - y6 e _{u Px sl - s4} -xl - x4 x2 + xs x3 + x6 Py 52 - ss - Y2 - Ys Yt + y4 y3 + y6

is therefore avoided in our problem •. In our computer programme2 6

we need only the following input data:

(1) the coordinates of the nuclei of the ions in the complex;

(2)

(3) (4) (~) (6)

the nuclear charge of each ion;

the basis functions (Slater-type orbitals) for each ion; the basls functions that can be fixed ("core" orbitals); the electronic donfiguration;

the \start vectors c~ and c~.

The part of the computer programme in which the atomic-orbital integrals (see chapter 4) are evaluated has been composed in such a way that identical integrals occurring in the problem are computed only once.

2. 5 SLATER-TYPE ORBITALS

The spatial part (denoted in spherical coordinates) of a basis function (atomic orbital)

x

is defined by:

(2.21)

in which Rnl (r) : the normalised radia"t part, i.e.

.. 2 2

of

r Rnl (r) dr = 17

Y lm ( e '4>) : the normalised angu"tar part7

n,l,m: the quantum numbers of the atomic orbital; r,e,!j>: the spherical coordinates.

The radial part Rnl (r) may be approximated by (a) a Slater-type function or (b) a Gaussian-type function. In our calculations we shall use a linear combination of Slater-type functions:

<;k1" 0

Several papers reveal that even a small number of terms will facilitate a good approximation of the Hartree-Fock atomic or-bitals. With the Gaussian-type function we would require more terms to obtain the same result.

Y1m(e,4>) are the normalised aomp"tex spherical harmonics defined by:

*

The indices n and 1 in Ckl,n and tkl will henceforth be avoid-ed in the expressions.

P₁ (cose) ~ ($)

m m {2.23)

and ~m ( ljl) (2.24)

The normalised associated Legendre functions P

1m(cose) are

de-fined by:

1 [21 + 1

~]

l

.

m

r

d ]1 +m 2 . 1

P₁m{cose)

=

2111 - 2- O+iii}T (-sine)

l""ii'Cci'S"6

(cos e-1) {2.25)

where -1 ~ m 5 1.

The normalised associated Legendre functions are related to the unnormalised ones by:

_ [21+1 (1-m)l

]l

plm {x) - - 2 - O+m)l plm (x) (2 .26) If m

=

_{O, P10}(x)

=

_P1 (x) becomes identical with the ordinary Legendre polynomials.

For these conventions the following identities are valid:

Pl _,-m (x) = (-1)m (1-m)l _{(l+m)l Plm} (x) (2.27a)

Pl (x)

,-m (-1)m plm (x) (2.27b)

•

(-1)m Yl (e,ljl) (2.27c)

Ylm(a,<Pl = _,-m

Throughout the present thesis these conventions of Rose27_{t for}

spherical harmonics are used.

The PeaZ angular functions s₁m(e,ljl) can be obtained by lin-ear combination of the complex spherical harmonics Y₁mce,ljl):

{Yl,-lml m slm (a,+) = N _m + _{(-1) ~~:m.Yl,lml}} (2.28 ) where K 1 m for m ~

o,

K m -1 for m <

o,

N i/12 for m <

o,

m (2.28a) Nm % for m=

o,

N _m

=

1/12 for m >

o.

t Note, the conventions of Rose2 7 are different from those of Margenau and Murphy,8 _{viz. Ylm (Margenau) • (-l)m Ylm (Rose),}

In this thesis the ket-notation of Dirac7 IX> will only be used for complex orbitals, while for real orbitals IX) will be used. This distinction has. been made to prevent misunderstandings in the expressions of integrals and accessory numerical data.

Those real orbitals that are essential for a discussion on transition-metal compounds, are assembled in table 2.3 which shows also their relation with the spherical harmonics.

Table 2.3 Ins)

lnp )

X

lnp )

_z

lnp )

_y

Essential orbitals for transition-metal complexes

- Rns 8

oo

= Rns 1

oo

- Rnp811 Rnp r!(Y1-1 - Rnp8 JO

=

R np - R _np Sl I _- = R np y10

I

nd 2 2) -X -y Rnd822 = Rnd

lnd )

_xz

I

ndz 2)

lnd )

_yz

lnd )

xy REFERENCES

1, Hartree, D.R., The Calculation of Atomic Structures, John Wiley and Sons, Inc., New York and Chapman and Hall, Ltd., London (1957).

2. Pople, J.A,, and Nesbet, R.K., J.Chem,Phys. 22, 57JL (1954). 3. LISwdin, P.-o., Ann.Acad.Reg,Sci.Upsalien. 2,127 (1958), 4. Nesbet, R.K., Revs,Modern Phys. 33, 28 (1961).

~.Kaplan, T.A., and Kleiner, W.H.,-phys.Rev. 156, l (1967). 6. Slater, J.C., Quantum Theory of Atomic StruCtUre,

McGraw-Hill, New York, 1960, Vol,I, ~hapter 9 and appendix 16;

Vol,II, chapter 17 and bibliography.

7. Dirac, P.A.M., Quantum Mechanics, Oxford (1947),

B. Margenau, H., Murphy, G.M., The Mathematics of Physics and

Chemistry, D. Van Nostrand Company, Princeton, New Yersey (1956).

9. Roothaan, C,C,J,, Revs.Modern Phys. 23, 69 (1951).

10, Roothaan, C,C,J,, Revs.Modern Phys.

32,

179 (1960).

12. L6wdin,

P.-o.,

Revs.Modern Phys. 32, 328 (1960).

13. Amos, A.T., Hall, G.G., Proc.Roy.SO'c. (London) 263A, 483 (1961).

14. Wilkinson, J,H., The Algebraic Eigenvalue Problem, Claren-don Press, Oxford (1965), chapter 5.

IS. Wilkinson, J,H., Numerische Mathematik~. 354 (1962),

16, Wilkinson, J,H., Rounding Errors in A1gebraic Processes, Notes on Applied Science, No. 32, H.M.s.o., London

(1963), chapter 3.

17. RC-Informatie no's 11, 13 (1967), Technische Hogeschool Eindhoven (unpublished),

18. L6wdin, P.-o., Advanc.Chem.Phys. 2, 207 (1959).

19, Amos, A.T., Mol.Phys. 5, 91 (1962).

20. Amos, A,T., Snyder, L.C., J.Chem.Phy.s. 41, 1773 (1964). 21. Sutcliffe, B.T., J,Chem,Phys. 39, 3322 (1963),

22. Cotton, F.A., Chemical Applications of Group Theory, John Wiley and Sons, Inc., New York, London, Sydney (1966). 23. Heine, V., Group Theory in Quantum Mechanics, Pergamon

Press, Oxford, London, New York and Paris (1960), 24. Hamermesh, M., Group Theory and its Application to Physical

Problems, Addison-Wesley Publishing Company, Inc., Reading, Mass., USA; London, England (1962),

25. Tinkham, M., Group Theory and Quantum Mechanics, McGraw-Hill Book Company, Inc., New York (1964).

26. DeLaat, F.L,M.A,H., Complete Set of Computer programmes for Unrestricted Hartree-Fock Calculations (ALGOL 60), Technische Bogeschool Eindhoven (1968), (unpublished). 27. Rose, M.E,, Elementary Theory of Angular Momemtum, John

3 SINGLE ANNIBILADON FOR A SINGLE DETERMINANT WAVE-FUNcriON

This chapter deals with the fact that the single

determi-nant wave function used in the UHF method will in general not be

a pure spin-state. To get an idea about this deviation some spin

P§Operties such as the average expectation value of the total

S spin-operator, and the spin density are considered before and

after spin pro~ection (single annihilation). Formulae will be

deduced for <S >, the charge-density and spin-density function.

3.1 AVERAGE EXPECTATION VALUE OF THE

s

2-0PERATOR

3.1.1 <Sa> before single annihilation

Prior to the evaluation of the average expectation va1ue of

the total

s

2 spin-operator after spin-projection, we shall

de-duce the expression of that quantity for the unrestricted HF wave function.

The average expectation value of the

s

2-operator, denoted as

a

<S >1 is defined as:

where Y

= '

(1,2, ••• , N) is the total wave function of a

N-elec-tron system (cf. eq. 2.2).

The

s

2 -operator can be written as:

N N

l s .• s . + 2

}:s .• s.

i • l l. l. i<j l. J

(3. 2)

The one-electron part in equation (3.1) becomes after

applica-tion of the orthonormality of the set of funcapplica-tions {~i}:

N

<'PI

I

s •• s.IT>

=

i•l 1 1

For the same reason the two-electron part yields: N

2<'1'1

L

s .. s.l'l'> i<j 1 J - <Tjl. (1)Tjl. (2)

ls

1

.s

2

1•·

(2}!p. (1)>] (3.4) 1 J 1 J

Substitution of the expression of s₁.s₂:

(3.5)

(3.6)

(for the

shift

operators: s~, s;,

s;,

and

s2,

see ref. 1) in

equation (3.4) yields:

N N

· 2<VI

L

s •• s.IV>

=\I

{{2a(m ,m

)-1]

i<j 1 J i<j si sj

z

.

Hence the complete expression for <S > is:

If there are in the N-electron system p a-spin electrons and q

electrons with ~-spin (assume p > q), the expression (3.8) can

be reduced to:

(3.9)

Substitution of the equations (2.13) and (2.15) into equation

(3.9) yields for the average value of the total

s

2 spin-operator

3

<8 >sd ~(p+q) + \(p-q) 2 - tr(PSQS) (3.10)

where tr is the trace of the matrix, and S the overlap matrix of the basis functions Xt•

For the special case of the restricted HF method we have ~~=~~'

~ ~

so that equation (3.10) can be simplified:

(3.11) in which s' stands for the spin of theN-electron system: s'

=

'J(p-q).

3.1.2 <83> after single annihilation

The average value for <83> after single annihilation As'+! (cf. section 2.3) is:

<_{8 2>} =

<Y'I8

2

I~'>

<~I I 'I''>

(3 .12)

We want to annihilate the component with a spin s = s' + 1 and assume that the intervention of spin components with s > s' + 1

may be neglected because of their much higher energy. A '+I

. 2 s

then be taken as 1.-dempotent, i.e. As'+l

=

As'+l. Moreover, commutes with As'+l so that equation (3.12) reduces to:

<'I'I821As'+l'l'>

<'flAs'+l'l'> (3.13)

We denoted <82> under these circumstances by <82> •

Substitu-asa

tion of the expression (2.20) for As'+l in equation- (3.13) yields:2 ' 3 <82> 1 4 _{(s'+1) (s'+2)<8}2> 8d}

=X

{-<S >sd + as a = <82>

-1

_{{<S4>sd + <S2>!d}} sd X (3 .14 ) where X

...

-<82> sd + (s'+1) (s'+2) (3.14a)

The expression <S2>sd (see ref. get: 5

for <S4>sd can be derived on the analogy of

4) and after substitution in equation (3.14) we

(3 .15 )

in which L

=

tr(PSQS). (3.15a)

Avoidance of the assumption A!'+l

=

As'+l in the beginning of this paragraph would have given a much more complicated ex-pression for <S2>. Amos and Snyderq have worked out this

expres-2

sion and denoted the corresponding expectation value for the S -operator by <S2> _{a a} • In the calculations figuring in chapter 6 of this thesis we shall use formula (3.15).

3.2 TOTAL ELECTRONIC ENERGY

The total wave function obtained by the UHF method will have a total

gies of the

electronic energy which is a mixture of the ener-components with a spins= s', s'+1, s'+2,... The energy of the components with s > s' is much higher than that of~ the component with a spins= s'. Thus, single annihilation will decrease the total electronic energy obtained by the UHF method. Again, using the assumption A:'+t

=

As'+l we arrive at an ex-pression for the electronic energy after single annihilation:

<E>

as a

<'l'jH!As'+I'!'>

=

<'l'jAs'+t'!'> (3 ,16) This expression is worked out in the paper of Amos and Hall3 _for

the extra assumption that the set of basis functions {xi} is orthonormal.

3.3 CHARGE-DENSITY AND SPIN-DENSITY FUNCTIONS

The aha~ge-denaity and spin-density functions denoted by

N

q

<;:>

<'I'

I

I

o <;:i ,;:>I'!'>

i=l (3.17a) N p(;:)

=

<'1'1

J

2Szio(;:i,;:) 1'1'>

1"'1 (3.17b)

where ;: stands for the spatial coordinates.

The equations (3.17a) and (3.17b) can be reduced by using the orthonormality of the set {1/J.}, and equations (2.13) and (2.15):

1

q (;:)

L

_{(P+Q) tu x:<;:>xu<;:>} (3 .18a) t ,u·

p(;:) =

_I

(P-Q) X (;:)X

•

(;:) (3 .18b) t ,u· tu t u

The corresponding expressions after single annihilation become (the derivationt of these equations can be found in refs. 5,6):

q(;:)

_I

(P'+Q') tu _xt<;:>xu(!:)

•

(3.19a) t,u p (~)

_L

(P'-Q')tu xt(;:)xu(;:)

•

(3.19b) t,u with P' p -

~{PSQSP-,(PSQ+QSP)}

(3. 20a) Q'

=

Q -

~{QSPSQ-,(QSP+PSQ)}

(3.20b) In chapter 6 we shall give graphs of these functions q(;:) and p(;:) for the TiF~- complex. We shall also calculate the total spin-density ps(;:F) at the F-site and from this the fraotiona~

density fs which is a measure for the isotropic hyperfine para-meter.7'8

REFERENCES

I, Griffith,

J.s.,

The Theory of Transition-Metal Ions, Cam-bridge Univ,Press, London and New York (1961}, p. 11.

2. L8wdin, P.-o., Phys.Rev. 97, 1509 (1955),

3. Amos, A.T., and Hall, G.G~ Proc.Roy,Soc. (London)~.

483 (1961}.

4. Amos, A.T., and Snyder, L.c., J.Chem.Phys. 41, 1773 (1964). 5, Sutcliffe, B.T., J.Chem.Phys. 39, 3322 (1963).

6, Amos, A.T., Mol.Phys. 5, 91 (1962).

7. Freeman, A.J., and Watson, R.E., Phys.Rev.Letters 6, 343

(1961).

4 MOLECULAR INTEGRALS WITH SLATER· TYPE FUNCTIONS

This chapter starts with the discussion of some basic

con-cepts and gives the complete formulae (in a few cases together

with their derivation) for the one- and centre one- and two-electron integrals (composed out of Slater-type basis functions)

as well as for the three-centre nuclear-attraction integrals.

For the two-centre exchange and the three-centre

nuclear-attrac-tion integrals a modified zeta-function expansion has been

plied, The three- and four-centre two-electron integrals are ap-proximated by several methods,

The last two sections deal with the relation between integrals

composed of complex functions and those composed of real

func-tions as well as the integral expression after an arbitrary

ro-tation of the coordinate axes.

4. 1 GENERAL CONCEPTS

The moteauZaP integPaZs in the unrestricted Hartree-Fock method (see section 2.2) will be composed in our case out of Stater-type basis functions, i.e. we use a multi-centre basis set. Corresponding to the number of different centres in the in-tegral we shall divide the set of integrals in one-, two-, three- and four-centre integrals. A subdivision can be made as to the number of electrons occurring in the integral (one- and two-electron integrals). A

lated to the unrestricted table 4.1.

survey of the molecular integrals re-Hartree-Fock method can be found in The following abbreviations have been chosen for the molecular integrals:

<AjMjB > -

f

x~A)*(1) M X ~B) (1} dv₁ ( 4.1 a)

~ _J

<AIMIA' >

_-

f

x~A)*<1> M x~A).(1) _{dv 1} (4 .1b)

~ _J

<ABjjCD> - Jfx~A)*(1) x~B)(1) ~ _J rl2 -I x<c> (2) xi D) • (2) dvk _{1dv 2} (4.1c)

where the index (A) represents the centre of the orbital x and

Table 4.1 Survey of the molecular integrals in the UHF method number of centres one: two: three: four: one-electron integral <AlA'> <AI-~ll lA' > <A 1-l" ~I I A f > <AjB> <AI-r;11A'> <Aj-~ll IB > <AI-r;11B > <AI

r~

1

1B

> two-electron integral <AA'IIA"A"' > <AA'II BB' > <AA'II A"B > <AB II A'B'> <AB II CC' > <AB II A'C > <AB II CD >

Before deriving analytical expressions for the molecular integrals in table 4.1 we introduce three concepts:

{1) the spheroidal coordinates; {2) the V _{n 1} {t,t)-functions, and

mp

(3) the Gaunt coefficients.

4.1.1 The spheroidal coordinates

A aharne distl"ibution _., O~~B) _{l.J - Xi} (A) (B) _{X j} for an electron is described in the Cartesian coordinate system by four independent parameters, for example xA, yA' zA and the internuclear distance RAB' some other possibility being the description with spherical coordinates. Both methods have the disadvantage of giving very complicated expressions after integration over the entire space. Introduction of the sphel"oidaZ coordinate system (see fig. 4.1) appears to have several advantages. The spheroidal coordinates

t, n and ~ are defined by (cf. ref. 1-3):

and conversely:

rB = l..iRAB(E:-n) rA = l..iRAB(F,;+n)

coseA = (1+E;n)/(E;+n) coseB = (1-E:n)/(E:-n) (4.2b) sinaA ={(E: 2;..1)(1 ...

n

2

)l~/(E:+n)

sinSB

~{(E:

2

;..1)~1

... n2)}1/(E:-n)

xs

Fig. 4.1 The spheroidal coordinate system

For integration in these coordinates, the volume element dv is (RAB/2)3(F,;2

-n

2)dE;dnd' and the integration limits for ' are: 0 to 21r, for n: -1 to 1, and for F,;: 1 to "'• The foci of this coordi-nate system coincide with the nuclei A and B which is very con-venient in applying numerical integration procedures.

The expressions for the Slater-type orbitals (STO's) on the centres A and B in these special coordinates can be found3 _by

substitution of the equations (2.23) and (4.2b) in:

= N. ~ in which Ni =

n.+!

(21;i) ~ { ( 2ni) I }

I

(B)

A similar formula can be derived for xj (E:,n,,).