A theoretical method for electronic structure calculations on systems of biological importance the group function approach

A THEORETICAL METHOD FOR ELECTRONIC

STRUCTURE CALCULATIONS ON SYSTEMS OF

BIOLOGICAL IMPORTANCE - THE GROUP FUNCTION

APPROACH

Donatella Paci

A Thesis Submitted for the Degree of PhD

at the

University of St Andrews

1994

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A T H E O R E T IC A L M E T H O D

F O R E L E C T R O N IC S T R U C T U R E C A L C U L A T IO N S O N S Y S T E M S O F B IO L O G IC A L IM P O R T A N C E

T H E G R O U P F U N C T IO N A P P R O A C H

-A Thesis Presented fo r the Degree o f D o c to r o f P hilosophy

in the F a culty o f Science of the U n ive rsity o f St. Andrew s

b y D o n a te lla Paci

ProQuest Number: 10167023

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D E C L A R A T IO N S

I, D o n a te lla Paci, hereby ce rtify th a t th is thesis has been composed by m yself, th a t it is a record o f m y ow n w ork, and th a t it has not been accepted in p a rtia l o r com plete fu lfilm e n t, o f any o th e r degree or professional q u a lifica tio n.

D o n a te lla Paci.

I was a d m itte d to the F a culty o f Science at the U n ive rsity o f St. Andrew s under O rdinance General No. 12 on 1st December 1988 and as a candidate fo r the Degree o f P h.D . on 1st Decem ber 1989.

D o n a te lla Paci.

I hereby c e rtify th a t the candidate has fu lfille d the conditions o f the R esolution and R egulations a p p ro p ria te to the Degree o f P h.D .

C O P Y R IG H T

In s u b m ittin g this thesis to the U n iv e rs ity o f S t.A ndrew s, I understand th a t I am g ivin g perm ission for it to be made available fo r use in accordance w ith the regulations o f the U n iv e rs ity L ib ra ry fo r the tim e b eing in force, subject to any copyright vested in the w o rk n o t being affected thereby. I also understand th a t the title and a bstract w ill be published, and th a t a copy o f the w o rk m ay be made and supplied to any bona fide lib ra ry o r research worker.

A C K N O W L E D G M E N T S

I w ould like to th a n k P rof. R. McW eeny fo r his invaluable assistance th ro u g h o u t the w o rk, and D r. C. Thom son fo r all his help d u rin g m y staying in St.Andrew s.

I w ish to acknowledge the Association fo r In te rn a tio n a l Cancer Research fo r p ro v id in g grants.

A B S T R A C T

T h eo re tica l m ethods fo r stu d yin g molecules o f biological im p o rta n ce are re­ viewed, b o th ab initio and sem i-em pirical. T he G roup F u nctio n A pproach is devel­ oped in d e ta il in its strong o rtho g on a l fo rm and corrections to the energy are added fo r ta k in g in to account n o n -o rth o g o n a lity effects, depending on the overlaps o f the group functions. A p p ro xim a tio n s are in tro d u ce d and tested so th a t this m e tho d can be applied to large molecules.

In p a rtic u la r, a system (o r a relevant fragm ent o f it ) is b u ilt up fro m localized tw o-electron groups, each one described by a two-electron group fu n c tio n (gem inal). Each group fu n ctio n is o p tim iz e d by using an SCF m ethod w ith an effective h a m ilto - n ian consisting o f the tw o-electron h a m ilto n ia n o f the group tog e the r w ith the effective p o te n tia l due to the presence o f the o ther electron groups (and to the e xternal e n vi­ ronm ent, eventually). T h e w avefunction fo r the whole molecule is an a ntisym m e trized p ro d u ct o f gem inals. T he energy is com puted as a sum of group co n trib u tio n s. C o r­ rections, depending on up to the second power o f the overlaps o f tw o groups at a tim e , are p a rtic u la rly im p o rta n t in co nfo rm a tio na l studies. The a pp ro xim a tio ns in tro d u ce d are based on the consideration th a t d ista n t groups consisting o f tw o p ositive and tw o negative charges see each o th e r cis n e u tra l e ntities and thus do n o t c o n trib u te appre­ cia b ly in the d e fin itio n o f the effective ham ilton ia n : the co m p u tin g e ffort is g re atly reduced in this way, the e rro r in tro d u ce d is sm all and can be estim a te d easily.

The theoretical m ethod presented in th is thesis offers a p o w e rfu l to o l fo r m a kin g q u a lita tiv e predictions o f th e changes re su ltin g from localized effects, such as tw is tin g around a bond, and it can be usefully applied to conform ational studies and geom etry o p tim iza tio n s. T he o th e r p ro pe rtie s w hich can be calculated are fo r the m ost p a rt d ire c tly related to the electron density; th is determines, fo r exam ple, the electrostatic p o te n tia l outside a m olecule and hence the position o f a tta ck by approaching ions o r p o la r species. C hem ical reactions, w hich involve breaking o r re-arrangem ent o f bonds, provide another vast fie ld o f a p p lica tio n. Such processes u sua lly involve o n ly localized regions in a m olecule and the adm ission of in tra g ro u p C l ensures th a t the stu d y o f bond breaking rem ains va lid th ro u g h o u t the whole process.

A T H E O R E T IC A L M E T H O D

F O R E L E C T R O N IC S T R U C T U R E C A L C U L A T IO N S O N S Y S T E M S O F B IO L O G IC A L IM P O R T A N C E

T H E G R O U P F U N C T IO N A P P R O A C H

-C O N T E N T S

P A R T I. T H E T H E O R E T IC A L T R E A T M E N T O F B IO M O L E C U L A R S Y S T E M S

5 L I In tro d u c tio n .

12 1.2 E m p iric a l E nergy C alculations.

15 1.3 Q u a n tu m C hem ical M ethods A p p lie d to the S tudy o f Large Molecules

o f B io log ica l Im portance.

1.3.1 A B IN IT IO M ethods.

1.3.2 S em iem pirical M ethods.

48 1.4 T h e Relevant Fragm ent and its E nviro nm e n t.

P A R T II. T H E G R O U P F U N C T IO N A P P R O A C H

64 , II. 1 T he G roup F u n ctio n A pproach: H isto rica l B ackground.

70 I I . 2 T he G ro up F u n ctio n A pproach: Present A p p lica tio n s.

75 11,3 P o ssib ility o f In c lu d in g N o n -o rth o g o n a lity Effects.

82 I I . 4 A p p ro xim a tio n s.

83 II.5 T he ’E ffective F ie ld ’.

87 I I . 6 Results and Discussion.

11.6.1 H ydrocarbons.

11.6.2 In tro d u c in g D ouble Bonds and Lone Pairs

11.6.3 A n A m in o A cid : Glycine.

11.6.4 Conclusions.

101 I I . 7 Perspectives.

105 A P P E N D IX A . H ydrogen-like O rb ita ls.

I l l A P P E N D IX B. O p tim iz a tio n o f A to m ic O rb ita l Basis Sets.

PART I

THE THEORETICAL TREATMENT

L I In tro d u c tio n .

T h e th e o re tica l tre a tm e n t o f b io m olecu la r systems is becom ing in cre asing ly im ­ p o rta n t in m odern science [1,2]. Such th e o re tica l studies a llo w in fo rm a tio n to be available th a t cannot be o b ta in e d b y e xpe rim en ta l m ethods. M oreover, w o rkin g h y­ potheses can be fo rm u la te d fo r th e e xpe rim e n ta list to test. T h e a im o f th is section is to review a num ber o f m ethods in use fo r s tu d yin g large system s, and to in tro d u ce the G ro u p F u n ctio n A p p ro a ch (developed in d etails in P a rt I I o f th is thesis) on w hich o u r a pp lica tio ns are based.

T h e s tu d y o f a chem ical system consists in lo o kin g fo r in fo rm a tio n about its s tru c tu re and its b eh a vio ur in presence o f o th e r chem ical system s (re a c tiv ity ). The m ost useful conceptual fra m e w o rk fo r the d escrip tio n o f chem ical s tru c tu re and reac­ tiv it y is represented b y P o te n tia l E nergy Surfaces (P E S ) [2a,3]. A P o te n tia l E nergy Surface is the m a th e m a tica l re p re sen tatio n o f the dependence o f th e to ta l energy Etot on the nuclear coordinates needed to define th e nuclear p osition s, assumed fixe d in space. F or a system o f M nuclei and N electrons, the to ta l energy is th e sum o f the k in e tic energies o f the nuclei AT(xn) plus th e k in e tic energies o f the electrons AT(xe) plus the p o te n tia l energy due to th e e le ctro sta tic in te ra c tio n between a ll th e particles (nuclei and electrons), U (x „ ,X e ),

E t o i = AT(Xn) 4- A'(X e) + C/(Xn,Xe) ( I . l . l )

I f we assume th a t the positions o f the nuclei are fixed in space*, the nuclear k in e tic energy is negligible and can be dropped fro m E q .( I .l.l) ; the system o f N electrons m oving in the fie ld o f M nuclei has a to ta l energy given b y

= A T ( X e ) + t / ( x „ , X e ) (1.1.2) w here Z7(xn,Xe) depends p a ra m e tric a lly on the nuclear coordinates { x ^ } . T h e value o f Etot in (1.1.2) is u ne q uivoca lly determ ined b y specifying the nuclear configuration. T h e nuclear co nfig u ra tio n can be specified in term s o f eithe r cartesian o r in te rn a l coordinates. In order to define the la tte r one has to m ention the m any experim ental evidences fo r the p o s s ib ility o f e x tra c tin g th ro u g h the stu d y o f a m o le cular system some lo ca l feature characterizing a sm aller system w h ich thus appears as a m ore or less d is to rte d co n stitu tive u n it. In th is way the concept o f bond was in tro du ce d and defined as a c o n s titu tive u n it localized between tw o neighbouring atom s. In te rn a l coordinates are given by b o n d lengths, bond angles and dihedral (o r to rsio n a l) an­ gles. A bon d length is the distance between tw o neighbouring atom s fo rm in g a bond, a bon d angle is the angle between tw o bonds w ith an atom in com m on and a d ih e d ra l angle is the angle between the planes form ed b y tw o pairs o f bonds having a bond in com m on. I f M nuclei are present, 3M -6 coordinates are needed in a p o lya to m ic m olecule to define the nuclear co n fig u ra tio n (3M -5 in a lin e a r m olecule). Therefore fo r a system o f M nuclei the P o te n tia l E nergy Surface has 3M -5 dim ensions (3 M -4 i f th e m o le cular system is lin e a r), since to the dimensions fo r nuclear coordinates

m ust be added the dim ension th a t measures the energy. T h e PES co nta in a large a m ount o f in fo rm a tio n a bo u t chem ical s tru ctu re and re a ctivity. Its m in im a represent (m eta )stab le nuclear co nfig u ra tion s (conform ations); its m a xim a and in fle c tio n p oints should be associated w ith unstable states and it is possible to stu d y the response to p e rtu rb a tio n (re a c tiv ity ) m o vin g away over the surface fro m a m in im u m . In fact a chem ical reaction can be th o u g h t o f as the changes a m olecule is subjected to when a p e rtu rb a tio n occurs. Such p e rtu rb a tio n can be due to the presence o f another m o­ lecule o r another p a rt o f the same m olecule or to an external fie ld . T h e s tru c tu re o f the m olecule in question, o rig in a lly in one o f its stable configurations, reactant (re­ presented by a m in im u m in a P E S ), changes continuously; it firs t assumes increasing values o f energy, u p to a m a x im u m o f energy corresponding to th e so called tra n s itio n state (represented by a saddle p o in t on the P E S ) and the n decreasing values o f energy dow n to a new stable co n fig u ra tio n (rea ctio n p ro d u ct). In p a rtic u la r, it is possible to evaluate a ro ta tio n a l b a rrie r in a m olecule. One speaks about in te rn a l ro ta tio n if some o f the d ihedral angles are changed w h ile o th e r in te rn a l coordinates rem a in unaffected. W h e n a (p a rt o f a) PES is determ ined as fu n c tio n o f certain selected d ih ed ra l angles, the energy difference betw een a neighbouring m in im u m and m a xim u m is called the b a rrie r to the corresponding ro ta tio n . T h e whole PES is u sually n o t know n; o n ly a p a rt o f it is studied. M a n y the o re tical m ethods are com m only a pp lie d to calculate the energy values. In any case, a series o f energy values are calculated correspon­ d in g to a p a rtic u la rly in te re stin g s tru c tu re and change; it is com m on p ra ctice to use m in im iz a tio n m ethods fo r fin d in g e q u ilib riu m structures.

T h e to ta l energy o f a m olecule is expressible in term s o f b o n d c o n trib u tio n s e^-, bon d -b on d in te ra ctio n s e*y, and so on. T h is conclusion follow s fro m the analysis o f a great num ber o f e xpe rim en ta l results and leads to a so-called e m p irica l bond system atic [4],

b o nd s bonds bonds bonds bonds bonds

E ^ o + E E E % ' ^ +

w here th e quantities 6^^,... do n ot depend on the environm ent. The very old em­ p iric a l system atics fo r the heat o f fo rm a tio n o f a molecule are examples o f num erical system atics. G ood results are o btained also if the czilculation is re stricte d to tw o-body in te ra ctio n s €,j between adjacent and vic in a l bonds. A n a ly tic a l system atics are also possible. They co n stitu te the basis fo r em pirical energy calculations which w ill be discussed later. We w ill see th a t o th e r m olecular properties are expressible in term s o f b o n d contributions. A n enorm ous body o f experim ental data can be interpreted by b o n d a d d itiv ity rules and sometimes by re la tive ly sm all deviations which in tu rn can be referred to special effects. Indeed one can id e n tify characteristic bond distances, b o n d angles, force constants, stre tchin g frequencies, in d ire ct spin-spin coupling con­ stan ts, etc.

For a given system m any in terestin g properties can be calculated by theoretical m ethods. T hey all are related to the changes in the electronic situ a tio n of the system , som etim es not affecting the nuclear configuration.

E le ctric properties are re la ted to the charge d is trib u tio n p{r) (charge density),

p (r) = - p e ( r ) + ^ ZAS{r - r ^ ) (1.1.3) A

w here P c(r) is the electronic d is trib u tio n ; Za is the atom ic num ber, is the p osition o f nucleus A , and

is the D irac delta fu n ctio n .

In te g ra tio n o f />c(r) over the whole space obviously gives the tota l num ber o f electrons (N ) in the m olecule,

T h e in fo rm a tio n contained in Pe(r) can be ve ry useful, fo r instance in m aking predictions on charge-controlled chemical reactions. One is often interested in the influence o f d iffe re n t atom s in the molecule and thus it is useful to approxim ate the charge d is trib u tio n in term s o f p o in t charges associated w ith the atom s fro m w hich the m olecule is b u ilt up. In this m anner the concept o f atom ic charge can be introduced. The a to m ic charge is not a physical quantity. I t can have many different theoretical d efinitions, re fle ctin g the a m b ig u ity of the concept. A conceptually sim ple d efin itio n is based on th e d ivisio n o f the whole space in to dom ains, one for each atom [5]. The a tom ic charge ça is then defined as

ÇA ^ Za - [ p e { r ) d V (1.1.4)

Jv^

where Va is the dom ain containing atom A. It is n ot always easy to fin d an appropriate d ivisio n o f space and the integrals in (1.1.4) are ra th e r d iffic u lt to evaluate.

Charge d is trib u tio n s can also be described in term s o f m ultipoles. The p ote ntia l energy E o f the charge d is trib u tio n Pe(r) in an e xternal field is

E = I Pc{r)V,r,{r)dV.

B y expanding V'ext(r) in a T a ylo r series, we have

B - e v . . . , ™ +

-where Q is th e to ta l charge o f the system, and r,- is x, y or z according to i = l , 2 or 3. T h e dipole m om ent has components pi given by

Pi —

J

Q i k = 5 y P e (r)[n rfc

-S im ila rly, hig he r m u ltip o le m om ents can be defined. T o ta l d ip o le m om ents can also be e stim a te d as v e c to ria l sums o f tra nsferable bond co n trib u tio n s to a s u rp risin g ly good accuracy.

A ve ry in fo rm a tiv e p ic tu re o f m olecular electrostatics emerges fro m an in te rn a l tra n s fo rm o f the m olecule charge d is trib u tio n , the M o le cular E le c tro s ta tic P o te n tia l (M E P ). T h e M E P U ( r ) th a t is created in the space around a m olecule b y its electrons and nuclei is given by

where is the e le ctron ic charge d en sity and Zm and Bure th e charge and p o s itio n ve cto r o f the m -th nucleus. F ( r ) m ay be regarded as th e p o te n tia l o f the m olecule fo r in te ra c tio n w ith an electric charge located at the p o in t r. T h u s a p o sitive charge is a ttra c te d to those regions in w h ich U ( r ) is negative since th is leads to a negative sta b iliz in g in te ra c tio n energy, and it is repelled fro m regions o f p o sitive p o te n tia l in w h ich the in te ra c tio n energy is destabilizing.

T h e accuracy o f in fo rm a tio n obtained fro m a ca lcu latio n is s tric tly dependent on the d e fin itio n o f a m odel fo r the system . T h is m odel is q uite sim ple fo r sm a ll systems in the gas phase and it contains o nly those n uclei and electrons w h ich d ire c tly p a rtic ip a te in th e process. In the m o d e llin g o f a b iom olecular s tru c tu re o r process the solvent m ust be included and a d is tin c tio n m ust be m ade between tw o b ro a d classes. In the firs t, p ra c tic a lly a ll atom s o f th e system c o n trib u te to the p ro p e rty o r take p a rt in the change, and therefore an o verall tre a tm e n t is necessary; e m p irica l energy ca lculations can be applied w ith some success. In o th e r cases, changes are locaHzed to a re la tiv e ly sm a ll p a rt, the relevant fragm ent. A num ber o f q ua n tu m m echanical m ethods has been used fo r s tu d yin g the la tte r. T h e p ro to ty p e approach is u s u a lly used, where

broken bonds at the fro n tie rs o f the fragm ent are sa turated by some a p p ro p ria te ly |

I

1.2 E m p irica l Energy C a lcu la tion s.

E m p iric a l m ethods are based on the observation th a t bond p ro pe rtie s are more o r less transferable w ith in ce rta in classes o f molecules. Regions o f p o te n tia l energ>* surfaces around th e ir local m in im a can be approxim ated by a sum o f sim ple analytical expressions w ith constants th a t are fitte d to reproduce a great num ber o f experim ental q u a n titie s as closely as possible [1,2b].

In the em pirical approach, the to ta l energy o f a m olecular system is approxim ated as

Etot = Efc + E t -f* Enh (1.2.1)

w here Eb, Et,Enh stand fo r b o n d in g, torsional and non bonding in te ra ctio n s respec­ tiv e ly .

B o n d in g interactions are a pp ro xim a te d by a sum of stretching, bending and cross te rm s

Eb = ^ 2 ■^‘■(^o) + ^ X / - ro )^ 4- ^ X ] - ^o)^ 4 cross terms (1.2.2)

j r 9

bending degrees o f freedom . Cross terms^ if considered, in clu de stretch-stretch, bend- bend o r bend-stretch c o n trib u tio n s. In general, these sim ple quadratic potentials are used, though m ore re a lis tic M orse potentials can be applied w ith little increase in com plexity.

The general fo rm o f th e torsional potential which accounts fo r the hindered ro­ ta tio n s around bonds is

= (1.2.3)

where is the m -fo ld ro ta tio n a l b a rrie r and 6 is the phase. The b a rrie r height is derived fro m the rates o f isom erization of molecules w h ile m and 6 are determ ined fro m sym m etry considerations.

The non b o n d in g te rm is usually expressed in the fo llo w in g general form

= e - ' + E C .,l(r ô ) " - (1.2.4)

The firs t te rm corresponds to electrostatic interactions between atom i and jf, bearing net charges o f qi and qj; e is the dielectric constant. T h e second te rm is the Hydrogen b o n d in g p o te n tia l. V an der Waals interactions are accounted fo r in the last te rm , where r*j = C ij and rjy are adjustable param eters, differing for various types of a to m ic pairs. In m ost cases n is equal to 12.

Several approaches are proposed which differ m a in ly in the param etrization.

The atom ic n et charges Çi are calculated by qua n tu m m echanical methods or derived together w ith th e Van der Waals parameters fro m the observed geom etry o f m olecular crystals o r fitte d to quantum -m echanically calculated electrostatic poten­ tia ls.

w here are adjustable param eters.

For exam ple, fo r am ide-carbonyl hydrogen bonds:

e

w here are fo r H ydrogen bon d in g in te ra ctio n s, A\j^B[j fo r V an der W aals in te ra c tio n s , is th e O — H separation, 9 is the H — N — O angle and a is an a d ju sta b le param eter.

1.3 Q u a n tu m C hem ical M ethods A pp lie d to the S tudy o f Large Molecules o f B io lo g ica l Im portance.

T he proper d e scrip tio n o f the electronic structure o f extended systems by state- o f-th e -a rt m ethods re m a in a challenging task o f quantum chem istry. These large system s are always b u ilt up fro m relatively sim ple b u ild in g blocks. The existence of such b u ild in g blocks o r fragm ents is a fundam ental p rin cip le in experim ental physical ch em istry [6]. The G ro u p F u nctio n T heory provides a the o re tical fram ew ork fo r the q u a n tu m m echanical tre a tm e n t o f this situation. It w ill be extensively discussed in the second p a rt o f th is thesis.

G enerally, those atom s d ire c tly involved in the chem ical process are treated quan­ tu m m echanically (rele va nt fra gm en t) and the rem ainder (fo r exam ple the solvent, or p a rts o f the m olecule fa r fro m the relevant fragm ent) are tre ate d as classical systems [2b]. M ore precisely, th e system can be fo rm a lly divided in to subsystems, one o f such subsystems being th e relevant fragm ent; the accuracy and the level o f description o f each subsystem decreases w ith the distance fro m the chem ically interesting fragm ent. The to ta l energy o f the system is the sum of the energies o f the subsystems plus the ir in teractio n s.

d e scrip tio n o f the system . I f a djusta ble param eters and e m p irica l q u a n titie s are in ­ tro d u ce d in the q u a n tu m m echanical ca lcu latio n , the so-called se m i-e m p irical m ethods are o bta in ed .

1.3.1 A B / m r / O m ethods.

1.3.1.1 T h e Schodinger E qu a tion .

T h e ele ctron ic s tru c tu re and pro pe rtie s o f any m olecule m a y be d eterm ined in p rin c ip le fro m the com plex fu n c tio n w h ich is a s o lu tio n o f S chodinger’s (tim e - in d e p e n d e n t) e quation [7]. For a system o f N electrons th is equ a tion takes the fo rm ,

Ê ^ ( X i , X 2, . . . , X a t ) = E ^ ( x i , X 2, . . . , X ; v ) (1.3.1.1)

w here x,- sym bolizes a ll the variables needed in re fe rrin g to e le ctron %,

x,‘ = (r,', y,‘, 0,’, S{),

Xi,yijZi sp a tia l variables, s,- spin variable (see A P P E N D IX A );

H is the H a m ilto n ia n o p e ra tor, associated w ith the observable energy, fo llo w in g fro m q u a n tu m m echanical p rin cip le , in a.u., *

Eh-fc(i) = - i v ^ ( 0 + V ( 0 (1.3.1.16)

— is the kin e tic energy o pe ra tor o f electron i;

V { i ) — — 7^ is the p o te n tia l energy o f electron i in the field of the nuclei n

w h ich are assumed fixed in space (B o rn-O ppenheim er app ro xim a tio n), r „ , being the m a g n itu d e o f the position vector o f electron i re la tive to nucleus n, and the charge o f nucleus n;

g { i , j ) = ^ is the electrostatic repulsion of electrons i and j , being the separation between electrons i and j ;

T h e fu n c tio n called the w avefunction, describes the state o f the N -electron system in the sense th a t can be in terprete d as being the p ro b a b ility o f fin d ­ in g electrons 1 ,2 ,..., TV in (space-spin) volum e elements d x i,d x2, r e s p e c t i v e l y (d r — d x id x2--.dxN)i provided ^ is norm alized, th a t is to say, f = 1.

G ive n this physical m eaning o f the w avefunction it follows th a t it must be co ntin ­ uous, single-valued and q u a d ra tica lly integrable; th is im plies th a t not every solution o f the Schrodinger equation is acceptable. Besides, P a u li’s P rin cip le states th a t the w avefunction m ust be antisym m etric.

< i4 > =

J

X'. = X l

Here  is the q u a n tu m m echanical (h e rm itia n ) o pe ra tor * associated w ith the observable A ; it w orks on fun ctio ns o f X i only. We w ill p u t x[ = X i afte r operating w ith À b u t before com pleting the in te g ra tio n

p i { x i ] x [ ) = N

J

is a generalized one-particle density fu n ctio n (the one-particle density m a trix ). In p a rtic u la r the diagonal elem ent p i( x i) = p i ( x i ; x i ) is the p ro b a b ility o f finding any o f the N electrons in d x i, independently o f the positions o f o th e r electrons. The ordinEir}' electron density fu n ctio n measured by X -ra y crystallographers, defining the electronic d is trib u tio n in space (pe in (1.1.3)), is obtained by in te g ra tin g p j( x j) over spin, P i( r i ) = J p i{x i)d s I. A tw o-particle density m a trix can be defined in the same way,

/3 2 (X i,X 2 ;x ',,X ;) = N ( N - 1) / ^ ( X l,X 2 , —, Xn) 9 ' ( Xj,X2, ...,XN)dX3...dXN-(1.3.1.26)

The energy E, in B orn-O ppenheim er app ro xim a tio n, is the energy o f the N elec­ tro n s m oving in the field provided by the nuclei. E assumes a set of values, cor­ responding to various electronic states. The to ta l energy o f the molecule, Etot, is o b ta in e d by adding the nuclear-nuclear repulsion energy to the electronic energy":

* A n h e rm itia n o p e ra to r À is defined fro m J <})*{À<j)j)dT = j [À(f>iY(j>jdr. It has re a l eigenvalues and orthogonal eigenfunctions, i.e. f 4>*<f>jdT = 0 if i ^ j .

E,ot = E + y ] ^ ^ (I.3 .:.3 )

Ba b being th e distance between nuclei A and B. E depends p a ra m e tric a lly on the nuclear coordinates. I t represents an effective P o te n tia l E nergy fo r th e nuclei, in the B orn-O p p en h eim e r a p p ro xim a tio n . In th is case the w avefunction fo r the w hole system (nuclei and electrons) can be w ritte n as a p ro d u ct = C ^ 2 (x n )^ (x i, X2, x / v r , P q), w here ^ represents the ele ctron ic p a rt o f the system in one o f its s ta tio n a ry states; it depends on the e q u ilib riu m nuclear co nfig u ra tion Rq. ^ J ( x „ ) is a m em ber o f a set o f nuclear w avefunctions o b ta in e d by solving the equation

one such equation fo r any electronic configuration. In o th e r w ords, Etot is used fo r d e fin in g p oints o f the P o te n tia l E nergy Surfaces fo r the given nuclear configuration, one fo r each electronic state.

1.3.1.2 T h e Independent P a rticle M odel. T he H artree-Fock Theory.

U p to now the essential principles o f qua n tu m ch em istry have been sum m arized. T h e y are the fo u n d a tio n o f th e o re tica l m ethods and a pp ro xim a tio ns used fo r descri­ b in g atom s and m olecules and the processes they are subjected to and co n stitu te the rem ote background o f the a pplications o f the G roup F u n ctio n approach we w ill develop in the second p a rt o f th is thesis, b u t they cannot be em ployed in calcula­ tio ns on systems o f great co m p le xity w ith o u t in tro d u c in g app ro xim a tio ns. In fact the S chrodinger equation is e xa ctly solvable o nly in a few cases. For the H ydrogen atom , fo r exam ple, one finds one-electron wavefunctions, called a to m ic sp in -o rb ita ls (S O ’s) (see A P P E N D IX A ).

one-electron h am ilto n ia n s H = - h(^i) and the corresponding w a ve fu n ction could be take n as a p ro d u ct o f one-electron eigenfunction, (a to m ic o r m olecular sp in o rb ita ls)

!

one fa c to r fo r each electron, I

I ' ï = i/’i(Xi)i/>2(x2)...i/'n (x n)- i I I In th is case we could easily o b ta in an exact so lu tion b y using the m e tho d o f se- j p a ra tio n o f variables and by solving a one-electron eigenvalue equation; th is is the j so-called Independent P a rticle M odel (IP M ). P roducts o f th is k in d are eigenfunctions | o f the h e rm itia n ope ra tor h and therefore the y fo rm a com plete set according to the | postulates o f q ua n tu m mechanics. Thus it is possible to describe the w avefunction ! ^ o f a m any-electron a to m o r m olecule as a lin e a r co m b ina tio n o f a ntisym m e trized {

pro du cts. I

A n a n tisym m e trize d p ro d u c t o f one-electron fun ctio ns can be constructed as a ! d e te rm in a n t b u ilt up fro m one-electron S O ’s {ipi}^ each one ’o ccup ie d ’ b y an electron, |

i.e., describing one electron, i

det

/ ^ i ( x i ) ^ i( x z ) . . . ^ i ( x ^ ) \ V '2 (x i) 1/'2(X2) . . . V'2(XN)

\V 'N (X i) ^2V(X2) . . . V ^ N (X ^ )/

(1.3.1.4a)

T h e d e te rm in a n ta l w avefunction (S later d e te rm in a n t) is often w ritte n in the sh ort­ hand n o ta tio n , \'ipixp2...i^N\ [S] and it is equivalent to the a n tisym m e trize d p ro d u ct

i[i/> i(x i)î/)2 (x 2 ). . . ^iv(xjv)] (1.3.1.46)

w here

 = cpP, (1.3.1.4c)

p

T h e Hartree-Fock m e th o d [9] uses these principles. T h e to ta l m any-electron w a ve fu n ctio n is w ritte n in the fo rm o f a single determ inant o f norm alized and orthog­ o n a l S O ’s.

1.3.1.3 The V a ria tio n M e th o d .

N o w we consider the p ro b le m o f o p tim izin g the SO ’s in the o ne-determ inant wave­ fu n c tio n . A fundam ental theorem states th a t the energy, calculated by m in im izin g th e fu n ctio n a l

o b ta in e d from the energy e xp e cta tio n value form ula E [^ ] = on replacing th e exact w avefunction 'I' w ith an approxim ate one containing variable param eters, p ro vid e s an upper bound to the tru e energj' (vn ria tio n a l p rin cip le ).

A system in its g round state is usually well described by a single determ inant of d o u b ly occupied o rb ita ls (closed-shell ground state); this means th a t the set o f SO’s used fo r the determ inant (occupied SO ’s) consists o f pairs o f S O ’s, the tw o o rb itals o f th e p a ir differing o n ly in the spin factor. We em ploy S later rules [8] to derive the e x p lic it fo rm of (1.3.1.4) in the case o f a norm alized one-determ inant wavefunction. T h e re s u lt is

E = ^ < 4>i\h\<i)i > ^ > (1.3.1.6)

» iJ

w h ere the sum m ations ru n over the occupied spin-orbitals and we have introduced th e ’’ antisym m etrized 2-electron in te g ra l”

< > = < 4>i<l>jWi(l>j > - < >

<

J

To o bta in a s ta tio n a ry value, we let -b Spk and set 6E = 0 (to first order) fo r all S(f>kj rem em bering th a t (1.3.1.6) remains valid only when the spin-orbiteds rem ain norm alized and orthogonal. W ith the change we obtain a corresponding firs t order va ria tio n o f the energy

= < 6(f)k\h\4>k > + C . C . + ^ ( ^ < Hk<f>)\\4>k<Pj > > ) + c.c.

j I

where each c.c. is the com plex conjugate o f the preceding term . The two sums in the parentheses, however, are identical, because the integrals are in va ria n t against exchange o f the indices on b o th sides of the ||, sim ultaneously, while the name o f the sum m ation in d ex (i o r j ) is im m ate ria l. Thus

6E^*‘^ = ( < S(f)k\h\<l)k > + ^ < S(j}k<l>i\\Ok4>i > ) + c.c. i

T he re su ltan t firs t order change 6E is SE =

I t is usual to in tro d u ce tw o new operators J, and A ',, defined as follows,

A ( l ) ( ^ ( X i ) = [

J g { l , 2 ) ( l ) i { X 2 ) ( l ) * { X 2 ) d X 2 ] < f > i X i )

I<i{l)(f){xi) = [

j

energy o f an electron at X i in th e fie ld due to a charge d is trib u tio n |<^,(x2)|^ (in

electrons per u n it volume). K i is the exchange operator.

I t is also convenient to in tro d u c e the to ta l coulomb and exchange operators, J = Ji and K = w here the sum m a tio n is over a ll occupied SO’s.

From these definitions o f J and K , it follows th a t the tw o-electron te rm in the energy expectation value (1.3.1.6) can be expressed in terms o f expectation values of a one-electron operator, thus

».J 3 3

= < 4> i\J - K \ ( f ) i > = < <f>i\G\(f>i >

where G = J — K.

T h e energy expression can be w ritte n as occ.spinorba

£ = E <<f:i\h + iG\<f>i>

I

F o r a closed-shell system, a fte r sp in in teg ra tio n occ.apac.orbs

E = ^ 2 < (pi\h 2 J — A )y% > = Yli 2 (^ ii + — A,-,) t

where y* is now the o rb ita l fa c to r o f the sp in -o rb ita l and

y /( r ) o = <^2/ - i( x )

y /(r)/? = (j>2iix)

ha = < <Pi\h\(pi > Jii = < y»jt7|yi > K ii = < y,-|A'|y,- >

D efining the Fock h a m ilto n ia n operator F = /i - f O , it results

= < 6y t | F | y t > + < y t|F |< ^ y t > = 0

Since this e v id e n tly gives the first order v a ria tio n o f a 1-electron energy expe ctatio n value €& = < y t | F | y t > , the Fock o p e ra to r m ay be regarded as the effective h am ilto n ia n fo r a single electron m o \in g in the fie ld o f the nuclei (contained in h) together w ith an effective " coulomb-exchange'' fie ld representing the presence o f a ll o th e r electrons.

T h e requirem ent th a t 6E^^^ be stationary stro n g ly suggests th a t the o rb ita l en­ ergies €k m ight be eigenv’alues o f F, b u t to ve rify th is conjecture we must take proper account o f the o rth o g o n a lity constraints,

< > = ^jk

where

^ L = I ^

^ 1 0 otherwise.

T h e new SO ’s y& - f S<pk m ust satisfy these co n d itio n , i.e. must be o rthog­ onal to ipk fo r p reserving norm alization and to every w ith j 0 for ensuring o rth o n o rm a lity in the new set o f orbitals,

< Sipkl'pk > + < ^k\Sipk > ~ 0

< <5y*|yj > = 0; < y ; |6y * > = 0; j k

< <^k\F\8ipk > - ^kj < y y l^ y t > = 0

w here ejk^^kj are a rb itra ry constants.

Since F is h e rm itia n ,

> = 0 V ;, A;

T h u s

^jk =

and

F ^ k - X ! = 0 VA; J

o r, in m a trix n o ta tio n ,

F ^ —# 6

w here $ = ( y i , y2, •••) and e is a square m a trix w ith elements h a vin g a rb itra ry values. A o n e -d e te rm in a n t w a ve fu n ction is in va ria n t u nder u n ita ry m ix in g o f th e o rb ita ls; new o rb ita ls # = # U m a y be in tro du ce d; and then

T hus, w ith o u t changing the m any electron w avefunction itse lf, we m ay always change the o rb ita ls so th a t a diagonal ë is obtained. D ro p p in g the bars,

In o th e r words, every o rb ita l satisfies = ey, and it is the n an eigenfunction o f an one-electron eigenvalue equation. Since F depends on $ th ro u g h the electron in te ra c tio n term s, the eigenvalue equation m ust be solved ite ra tive ly. T h is is the Self C onsistent F ie ld (S C F) m e tho d fo r solving an H artree Fock equation.

E ve ry tim e an eigenvalue equation has to be solved where the d e fin itio n o f the o p e ra to r is dependent on the fo rm o f the w avefunctions to be o p tim ize d a SCF m ethod can be em ployed. T h is is the case, fo r exam ple, o f the G roup F u n c tio n M e th o d ; fo l­ lo w in g fro m the d ire ct a p p lica tio n o f the v a ria tio n a l theorem to a v a ria tio n a l wave­ fu n c tio n defined as an a ntisym m e trized p ro d u ct o f (a n tisym m e tric) g roup wavefunc­ tio n s, an eigenvalue equation rem ains to be solved fo r each electron g roup, having an effective h a m ilto n ia n defined as a fu n c tio n o f the w avefunctions o f the o th e r groups and describing the effective fie ld the y generate around the electrons o f th e group in question. A great n um ber o f the o re tical procedures are in fa ct a pp lica tio ns of the SC F m ethod.

Reasonable firs t a pp ro xim a tio ns to the a to m ic o rb ita ls fo r heavier atom s are hydrogen-like o rb itals, o b ta in ed fro m the H ydrogen o rb ita ls b y s u b s titu tin g th e ir exponents w ith variable param eters. L in e a r com binations o f S later o rb ita ls o r gaus- sian o rb ita ls are often used to reproduce a to m ic o rb ita ls (see A P P E N D IX A ). T he SC F m e th o d has been extensively applied to calculations on m olecules too after the in tro d u c tio n o f the R oothaan L C A O (L in e a r C o m b in atio n o f A to m ic O rb ita ls) ap­ p ro x im a tio n fo r describing one-electron m olecular w avefunctions (M o le cu la r O rb ita ls, M O s) [10].

In general it is always possible to expand each y , as a lin e a r co m b ina tio n o f

y : = = XT 3

— $ € becomes, fo r a closed-shell system,

F T = eST

where

F fil/ — hfn, -f- G fif/

Sfit/ = < XmIx*' ^

hfii/ = < ^

= E > ,„ ^ a K /ji^ IA c t) - K/JC'lAy)]; (/ii/|A<r) = < ^A1^ |i/<7 >

occ

F<ta — 2 ^ TffjT^j 3

I

1.3.1.4 The E le ctro n Density. |

A to m ic and m olecular states can be described generally by using lin e ar com bi­ n a tio n s o f determ inants o f spin-orbitals, the Hartree-Fock w avefunction being sim ply a o ne-determ inant a pp ro xim a tio n. Since each term is defined by sta tin g a ’’ configu­ ra tio n ” o f occupied SO ’s, this expansion procedure is often referred to as the m ethod o f ’’ C o n fig u ra tio n In te ra ctio n ” (C l). For a state described by a 'F w hich is a linear co m b in a tio n o f S later determ inants the density fu n ctio n can be expanded in the form

p ( x i; x i) = X ^ p R S 0 /t(x i)V ’J ( x i) (1.3.1.7a) R,s

T h e one-electron energy becomes,

< X ] ^ (0 > = X / ^ > .

i R,S

T h e re q uired pRs is thus s im p ly the coefficient o f < > in the usual o rb ita l expression fo r th e one-electron energy. T h e d ensity m a trix w ith o u t reference to spin is s im ila rly

= X I PR S ^^R ^ri ) y g ( r i ) (1 .3 .1 .7 6 ) R,S

w here Pr s is a n um erical coefficient obtained as the result o f spin in te g ra tio n . I f we

express each one-electron fu n c tio n as a lin e ar co m b in a tio n o f a rb itra ry basis fun ctio ns X = ( x i) X 2, - ” jX m ), th e n y i t ( r i ) = and the e le ctron d e n sity takes the

general appearance

-P (n ) = P ( r i ; r i ) =

= P R sTm T-s R,S

F o r a closed-shell system Pr s = 26r s and PX becomes P ^x as defined in the

pre viou s section.

W h e n the basis consists o f o rb ita ls (e.g. A O ’s) w ith a w e ll-de fin ed lo ca liza tio n in space th e coefficients p e rm it a q u a n tita tiv e analysis o f the fo rm o f th e electron d is tri­ b u tio n . W e define n orm a lize d o rb ita ls and overlap densities (assum ing fo r convenience real o rb ita ls )

d%;(r) = % i( r ) x ;( r ) /5 ':;, overlap density, w ith S ij = < X i ( ^ ) \ X j i ^ ) > , and fin d

^ ( r ) = X Z + X Z W (L3.1.8)

* %<;

w here qi = P a, qij — 2 S i j P i j , {i j ) .

Since in te g ra tio n o f P ( r ) over all space m ust y ie ld N (th e n u m b e r o f electrons)

it follo w s th a t

N i<3

T h e to ta l charge is d ivid e d o ut in such a w ay th a t an a m o u n t qi arises fro m each o rb ita l density d{ and qij fro m each overlap den sity dij. These q u a n titie s, firs t proposed b y M cW eeny, are usually referred to as o rb ita l and overlap populations. T h e idea o f p o p u la tio n analysis has been extensively developed b y M u llik e n [11] b u t th e p o p u la tio n s assigned to the various regions o f space are n o t unique, depending e n tire ly on the choice o f th e basis o rb itals; it is therefore clear th a t the n ature o f chem ical bonds cannot be studied sim p ly fro m an in sp e ctio n o f b o n d p op u la tion s.

1.3.1.5 Valence B on d m ethod.

T h e H P m odel is n ot apt to describe the s p littin g o f a m olecule in to tw o parts when a bond is broken. T h e re su ltin g pieces, in fa c t, are open-shell system s and are therefore b a d ly described by single determ inants o f d o u b ly occupied o rb ita ls.

A b e tte r tre a tm e n t o f this phenom enon is given by the Valence B on d theory where each electron occupies an atom ic o rb ita l and each chem ical b o n d is associated w ith th e p a irin g o f the spins of two-electrons in the valence o rb ita ls o f the atom s they belong to ; fo r a p o ly a to m ic m olecule w ith N sin g ly occupied valence a to m ic o rb ita ls, y i , y 2, the o rb ita l p ro d u ct fu n ctio n y i ( x i ) y 2( x2) . . . y ^ ( x ^ ) , is com bined w ith

single o r p a ra lle l coupled electron. T he P au li p rin cip le is satisfied a p p ly in g the usual a n tisym m e trize r  = Y2p e p P defined in E q.I.3.1.4c.

W ith neglect o f overlap one m ay define

Q — < y i y 2 . . . y N | F | y i y 2 . . . y N >

and

Kij

yiy2---yi---yi-.-yiv|F|yiy2-.-yi---yi---yiv

E = Q + /- C .- y - i 5 3 K i j - 5 3 K i j (I.3.1.9)

( i j ) p a i r e d ( i j ) u n c o u p l e d { i j ) p a r a l l e l

T h is fo rm u la gives the "p e rfe ct a irin g ” a pp ro xim a tio n, using one electronic con­ fig u ra tio n corresponding to the spin coupling scheme associated w ith a h yp o th e tica l a llo ca tio n o f chem ical bonds. W h e n a single co nfig u ra tion does n o t give an adequate p ic tu re o f the bonding, a ll the plausible stru cture s are a d m itte d w ith various p a irin g schemes and the w avefunction is represented as a lin e ar co m b ina tio n. T h e o p tim a l m ix in g coefficients are obta in ed , as usual, by solving a m a trix secular equation. A t large distance, afte r b reaking a bond, a correct w avefunction is o bta in ed , describing isolated pieces o f the m olecule.

T h e m a in d iffic u lty o f th is m e tho d consists in ta kin g in to account o rb ita l over­ lap, w h ich g re a tly increases the m a th e m a tica l com plexity. T h is is w h y the m ethod has been displaced w ith the developm ent o f ab in itio m ethods o f co m p u ta tio n . Se­ veral form s o f the th e o ry are now available, able to face th is p ro ble m and in d ic a tin g p ro m isin g developm ents.

1.3.1.6 C o rre la tio n.

due to th e a n tis y m m e try o f the w avefunction, b u t it is n o t considered fo r electrons w ith opposite spins and in tro du ce s an e rror o f ro u g h ly one per cent; u n fo rtu n a te ly , th is corresponds to energy differences com parable, fo r exam ple, w ith m o le cu la r b in d ­ in g energies, b a rrie r heights, m olecular in te ra c tio n energies th a t the th e o ry should be able to p re d ict. T h e co rre la tio n energy is usua lly defined as th e difference between the exact energy and the best o ne-determ inant to ta l energy.

T h e greatest efforts have been em ployed by the o re tical chem ists to stu d y the co rre la tio n problem .

T h re e m a in m ethods have been applied:

(i) one can determ ine th e coefficients o f a C l expansion d ire ctly, along w ith the firs t few energy levels, b y d ire ct solution o f the secular e q u a tion , using the m ost p o w e rfu l tecniques available fo r evaluating m a trix elements and so lvin g the m a trix eigenvalue problem ;

( ii) long C l expansions can also be handled by s ta rtin g fro m a leading te rm 0Q, w h ich w o u ld be an exact eigenfunction o f a m odel h a m ilto n ia n Hq, and re g ardin g

H — H q as a p e rtu rb a tio n u nd e r whose influence the rem a in in g C l coefficients w ill as­

sume non-zero values, w h ich can be estim ated b y m a n y-b od y p e rtu rb a tio n th e o ry [12], using d ia g ra m m a tic tecniques, w ith o u t a c tu a lly solving secular equations. R ayleigh- S chrodinger p e rtu rb a tio n th e o ry [7], where the energy is expanded in term s o f sum - over-states form ulae, fo rm s th e basis o f m a n y-b od y p e rtu rb a tio n th e o ry ; ca nce lla tio n o f u nphysical, n o n -lin e a r term s in the sum -over-the states form u la e is a fea tu re o f m a n y -b o d y fo rm a lism w ith in the algebraic a pp ro xim a tio n , i.e., b y using fin ite ana­ ly tic basis sets. T he M ^lle r-P le sset p e rtu rb a tio n th e o ry [13], is e ssentially one fo rm o f th is p e rtu rb a tio n approach and is now w id e ly used. T a kin g a sin g le -d e te rm in a n t

^ 0 as reference fu n c tio n fo r the C l expansion ^ = #o 4- Y2 th e re m a in in g

x > o

single excitations can be e lim in a te d ; fo r electrons double e xcitations are o f the great­ est im portance, h igher-order corrections to the w avefunction being well expressed as antisym m etrized p ro d u cts o f th e m . Such an approach leads to a C orrelated E lectron P a ir A p p ro xim a tio n , C E P A [14], o f w hich there are m any V2irietes; if higher-order corrections are neglected, i.e. the electron pairs are considered com pletely indepen­ dent, an Independent E le c tro n P a ir A pp ro xim a tio n , lE P A [15], results in w hich the co rre la tion energy, at least fo rm a lly , can always be expressed as a sum o f electron p a ir co ntribu tio ns. A separate S chrodinger equation is then solved fo r each electron pair, and in firs t a pp ro xim a tio n th e re su ltin g p a ir correlation energies are sim ply summed to give an estim ate o f the to ta l co rre la tion energy.

( iii) A system can be fo rm a lly d ivided in to weakly in te ra ctin g subsystems, cor­ re la tio n being calculated b y C l w ith in each subsystem (G ro u p Function Approach). T h is m ethod is conceptually d iffe re n t from the other illu s tra te d in (i) and (ii) be­ cause it stresses the im p o rta n ce o f local co n trib u tio n s to the properties of the whole m olecule. The group w avefunctions, each one describing a group o f electrons, are u sua lly b u ilt up by m a kin g an a p rio ri assum ption o f lo ca liza tion o f the electrons. For exam ple, we can suppose th a t two-electron groups are able to represent bonds. I f every bond o rb ita l is b u ilt as a lin e ar com bination o f tw o one-electron wavefunc­ tions, a bonding ( r i) and an a n ti-b o n d in g ( t2) o rb ita l, then the a ntisym m etric group fu n c tio n can be defined as

+ A ^ ||r2f2||)

w here N r is a n o rm a liza tio n fa cto r. The to ta l w avefunction is an antisym m etric p ro d u c t of group fun ctio ns

’F = Â [0 ^ $ g . . .0 A...]

— ^ a ) + A ^ (rg r2( a ^ — ^Of))]

^ = c ° {Â ^ [a ia i6i6i.. . r ir i.. . ] + V ' ^ ,^ Q ia i6i6^ .. 1 1

“ n r i

T T + _ }

R ^ n n s i s i

In th is case the antisym m etrizer is À' ~ Y lp ^ p E where P is any p e rm u ta tio n between the variables and c® = first te rm in is a S later dete rm ina n t

b u ilt up fro m the one electron functions { r i } , the rem aining term s are wavefunc­ tions o b ta in e d fro m ^ on replacing r i f \ by r2T2, i.e., by double su b stitu tio n s, o r by qua d ru ple su b s titu tio n s and so on. and the y give rise to the correlation energy.

1.3.1.7 Localized O rbitals.

M o le cu la r ca lcu latio n s would stand to gain considerable in tu itiv e appeal if they contained some d ire c t recognition o f localized bonds [4,2c, 16].

L o ca l b o n d fu n ctio n s were n a tu ra lly employed in the valence-bond the o ry of m o le cular e le ctro n ic stru ctu re [6]: each bon d was described as a tw o-electron closed- shell system w ith an antisym m etric spin fun ctio n, — ^ a ) and a sym m etric sp a tia l fu n c tio n assumed as a linear com bination o f aa, bb and ^ ( n 6 -f 6a), a and b being a to m ic w avefunctions for the two atom s o f the bond respectively.

share th e p e rm u ta tio n p ro pe rtie s o f bonds, being inconverted b y the sym m e try ope­ ra tio n s o f the m olecular p o in t group, and are h ig h ly localized in the bon d regions. Lennard-Jones and P ople suggested th a t, since equivalent o rb ita ls are h ig h ly lo ca li­ zed, an electron occupying one o f them w ould have m a xim u m C oulom b in te ra ctio n w ith th e electron sharing th a t o rb ita l . E dm iston and Ruedenberg used th is idea and devised an ite ra tiv e n u m erical m ethod fo r im p le m e n tin g th is self-energy lo ca liza tion c rite rio n also fo r cases where sym m e try equivalence could n o t be defined [18]. A steepest-ascents m e tho d has also been suggested [19]. Foster and Boys th o u g h t o f localized o rb ita ls w h ich m in in rize the sum o f separations between o rb ita l centroids o f charge [20]. T h is ty p e o f lo ca liza tio n has been called an ’’in trin s ic lo ca liza tio n ” c rite rio n since it in no w ay it depends on the L C A O basis. A n o th e r in trin s ic c rite rio n is due to von Niessen. T h e vo n Niessen or D ensity c rite rio n makes use o f the electron den sity ope ra tor m a xim izin g th e self-overlaps o f the L M O densities [21].

density m a trix [27]. P ro je c tio n m ethods give a set o f quasi localized o rb ita ls. T he p rin c ip le is as follows: a set o f ad hoc com pletely localized o rb ita ls is p ro je cte d in to the space spanned by the p rim itiv e delocalized o rb ita ls, and the re s u ltin g o rb ita ls are reorthogonalized by the L o w d in tra n sfo rm a tio n , g ivin g the fin a l set o f quasi-localized o rb ita ls [28].

T h e d ire ct ca lcu latio n o f local o rb ita ls has been discussed b y E d m in sto n and Rue­ denberg [18] and in m ore details b y G ilb e rt [29], A dam s [30], Peters [31], W ilh itte and W h itte n [32], D audey [33], Letch er and D u n n in g [34], A nderson [35], vo n Niessen [36], H uzinaga [37]. There are also the so called ’bond fu n c tio n s ’ w h ich are n o t centered on any o f the atom s, b u t are placed between tw o neighbouring ones to represent bonds [38]. These d ire ct lo ca liza tio n schemes, tho u gh the y are c e rta in ly ve ry im p o rta n t fro m the th e o re tica l p o in t o f view , d id n o t help too m uch in p ra c tic a l applications.

T h e concept o f localized o rb ita ls goes fa r beyond the fra m e w o rk o f the pure H F th e o ry: it can also be used in connection w ith correlated fu n ctio n s [39,33].

transform ed in to o rb ita ls w h ich are directed towards two-centre bonds o r lone pairs.

P a u lin g form ulated the firs t d e fin itio n s o f h yb rid o rb itals (HA O s) [6). His analysis o f th e bond revealed th a t the bond is stronger if the concentration o f o rb ita ls in the in te m u cle a r region is greater and he was the first who discussed the prin cip le o f m a xim u m overlapping. However, instead o f calculating overlap integrals, he defined the b o n d strengths of th e o rb ita l as the m axim um value o f the angular p a rt of the o rb ita l fu n ctio n norm alized to 4 7t over the surface o f a sphere.

Since the 1-q u a n tiza tio n does n o t hold anymore in p e rtu rb e d atom s embedded in th e in tra m o lecu la r fie ld , one can freely m ix s, p, d, f... A O ’s. Assum ing h ybrid o rth o n o rm a lity and em ploying a m a xim u m bond strength c rite rio n , equivalent te tra ­ h ed ra l sp^ hybrids are easily constructed:

XI = § [(n s) -b ( n p i) 4- {npy) + (n p .)l X2 = |[( n s ) 4- ( % ) - {npy) - (n p ^)]

X3 = |[(n s ) - (n p x) -f (npj,) - {np,)] X4 = |[( n s ) - (n p x) - {npy) 4- (n p ^)]

P au lin g observed th a t the degree o f h ybridizatio n , when it is n o t determ ined p u re ly by sym m etry, fo r exam ple in the presence o f lone pairs, is stro n g ly dependent on th e (n s)-(np ) energy gap, being m ost effective fo r the firs t row atom s where the gaps are small.

F o r Ein h yb rid o rb ita l the s-character can be defined as

(.% )

c2(2s) 4- c2 (2 p i) 4- c2(2py) 4- c - ( 2 p j

Interestingly, P au lin g explained double and trip le bonds by tw o tetrah e dra shar­ in g a common edge and side, respectively [40], i.e., by tw o bent bonds, where the e lectron density is concentrated in tw o regions o f space more d ista n t fro m each o th e r th a n in a (7 — 7T description. T h e stru ctu re w ith bent bonds can be, therefore, the energetically favoured.

D uffey and co-workers used the c rite rio n o f m axim um a m p litu d e in stu d yin g a n um ber o f large molecules [44].

A p a rt from the m a xim u m a m p litu d e crite rio n , there are m any other ways o f con­ s tru c tin g local hybrids A O ’s. One o f them is the Localized A to m ic O rb ita l (L O A ) m e th o d designed by A ufderheide and C hung-P hillips [45]. They m axim ize the fun c­ tio n La fo r each atom A , where

A occ occ

-a

and denotes the conventional bond order m a trix. The basis set o rb ita ls w h ich yie ld m axim al value o f L a are local orthogonal h yb rid o rb itals. In a ca lcu latio n on H2O tw o ’ra b b it-e a r’ lone pairs result; and the lone pairs hybrids are found to have a h ig he r s-content.

In the Generalized Valence B on d (G V B ) m ethod, h yb rid iza tio n follows as a conse­ quence o f the fu ll va ria tio n a l ca lcu la tio n and is not introduced as an a p rio ri postulate. E ssentially the G V B w avefunction can be viewed as generalization o f a closed-shell H F fu n c tio n  ('0 iî^ iî/j2'0 2- - ) where À represents a custom ary antisym m etrizer, in w hich each p a ir of o rb itals is replaced by {^iaK'ib + ^ib'^ia)ct0 where the firs t p osition in p roducts im plies electron 1 and the second electron 2.

A large num ber o f molecules have been treated by the G V B procedure and the corresponding hybrids are available [46]. U sing this m ethod Messmer et al. have show n th a t a bent bond p ictu re describes m u ltip le bonds in C O2, C2P4, and C2H2

tures. Inclusion o f co rre la tion between pairs o f coupled electrons makes the bent bond p ictu re even m ore accurate [47]. We shall use a s im ila r m odel w ith in the G roup Func­ tio n form alism in the second p a rt o f the present w ork. The localized o rb ita l p icture retains its v a lid ity even in archetypal delocalized molecules like benzene, p yrid in e and the related six-m em bered rings [48]. It appears th a t arom atic s ta b ility arises more fro m the mode o f spin co up lin g th a n fro m o rb ita l delocalization.

A no th er way o f e x tra c tin g h y b rid iz a tio n param eters fro m m olecular wavefunc­ tio ns is by m aking use o f the firs t o rder density m a trix . L et us first consider the early p re scrip tion o f M cW eeny and D el Re [49].

S ta rtin g w ith A O ’s a fte r L o w d in ortho g on a liza tio n, the localized bond form ed b y coupled h ybrids o rb ita ls X r and x» takes the fo rm cos 6rsXr + sin^raXa, where 6rs is the ionic-character param eter. M cW eeny-D el Re m ethod im plies a u n ita ry tra n sfo rm a tio n o f in ita l pure A O ’s and subsequent SCF calculations. A m a trix P can be defined

(

:

1 -f cos 2^, sin 20t

sin 29r3 . . . 1 — cos 2â,

t . . . . . . . . . . . . ... 2 J

where Pit = 2 describes the lone p a ir corresponding to tw o coinciding hybrids. If Qra ~ 45® one has a p u re ly covalent bond where Prr — Psa = Pra — Par = 1. In general, neith e r the h yb rid s are com pletely localized n o r the bonds are 1 0 0% covalent. However H A O ’s can be constrained to satisfy one o r b o th o f these conditions leading to an increase in to ta l energy o f a molecule. In th is way effects o f io n ic ity and delocalization can be e xtim a te d .

h y b rid o rb ita ls [50]. S im ilarly, if one diagonalize extended su b m a trix associated w ith b on d ed atom s, one obtains eigenfunctions describing bond o rb itals.

Inste a d o f diagonalizing the density m a trix P , a stra ig h tfo rw a rd calculation of h y b rid s- and p-character is possible [51]. T he idea was based on an early observation o f W ib e rg th a t the square o f the bond order in CH bonds is related to th e ir s-character. T h e b o n d charge can be expressed as:

T h e b o n d in g charge o f all fo u r 2s and 2p A O ’s o f the atom A ta kin g p art in the A — B bond is D efining Wg(^AB) ^ the p a rt o f the 2sa o rb ita l involved in the A — B covalent bond W^(^ab) = P2sa„^2sau^ the s-character o f the x a b h yb rid is given by:

{ l / 2 ) W ^ ^

T h e firs t general m ethod fo r co nstru ctin g h yb rid o rb itals used the m axim um o verlap co n d itio n . M a xim u m overlap m olecular o rb ita ls (M O M O ) received a detailed d e scrip tio n a nd this m ethod is easily im plem ented even in largest systems.

D e l Re discussed a way o f o b ta in in g h y b rid o rb ita ls A O ’s w ith in the fram ework o f th e effective one-electron ham ilton ia n where the wavefunctions are expressed in a fo rm o f lin e a r com bination o f atom ic o rb ita ls, solving by a u n ita ry transform ation of the p u re A O ’s so th a t the overlap m a trix has the pseudo-diagonal 2 x 2 form as close as possible [52], the equation

H C = S C Ê

w here 5 jj = < ^i|x> > , being a lig a nd o rb ita l in a system o f the type ...rjt) ^ and X j being an o rb ita l om atom M .

o rb ita ls

X» = ai(2s) + (1 - a,)^/^(2p)<

is varied u n til the sura o f bond energies is m axim ized.

A plausible assum ption th a t the bond energies are lin e ar fun ctio ns o f the cor­ responding overlap in teg ra ls Ea b — ^a bSab 4- Ia b is made, where Ua b and Ia b are e m p iric a lly adjusted param eters. The hybridizatio n indices a,- are subject to o r­ th o n o rm a lity conditions

aiGj 4- (1 — cosOij = <5,y

fo r z, j = 1...4, w here 6ij is the angle between the a xia l sym m etric h ybrids tpi and x})j sharing the same atom . D u rin g the calculations the H A O ’s are allowed to follow the directions o f the s tra ig h t hnes passing through the d ire c tly bonded nuclei. To be q u ite precise, bond angles are determ ined by the in te rh yb rid angles o f the o p tim a l h yb rid s. O rth o g o n a lity and perfect o rb ita l follow ing o f h yb rid s are not com pletely ju s tifie d suppositions b u t the y w ork very well in most cases. T h e y considerably add to the sim p lic ity o f the m e tho d .

F in a lly, in te ra to m ic overlap integrals are forced to follo w a predeterm ined linear b o n d o ve rla p /b o n d distance correlations.

In the firs t ite ra tiv e step the in itia l bond distances are selected b y an educated guess and the independent h y b rid iz a tio n parameters a, are varied u n til the m axim um o f Eh is achieved. T h e n the n ext set o f bond distances is deduced and the h y b rid iz a tio n indices are reoptim ized. T h e w hole ite ra tive procedure is repeated u n til a consistency between the in p u t and th e o u tp u t bond distances is extablished. Hence, the IM O scheme is a constrained w eighted m axim um overlap m ethod.

1.3.2 S em iem pirical Developm ents.