E L E C T R O N S T R U C T U R E O F

O R G A N O P H O S P H O R U S C O M P O U N D S

I . THEORY AND PARAMETRIZATION OF THE ITERATIVE

METHOD OF COMPUTATION, ALLOWING FOR A L L THE

VALENCE ELECTRONS

V. Y a . A r t y u k h o v a n d V. I . D a n i l o v a UDC 539.192.194

The theory and p a r a m e t r i z a t i o n of an i t e ra t ive method of calculat ing the e lec t ron s t r u c t u r e of molecu les a r e p r e sen t ed . Use is made of the modif ied Mulliken equation for calculat ing the populat ion of the a tomic orb i t s , while group theory is employed for es t imat ing the p a r a m - e t e r s . A c o m p a r i s o n between the ca lcula ted and exper imenta l ionizat ion potent ia ls of c e r - ta in molecu les conf i rms that the s i m p le computing method h e r e p roposed yields r easonab ly s a t i s f a c t o r y r e su l t s .

1 . I n t r o d u c t i o n

In this p a p e r we shal l se t out a method of calcula t ion which we have r ecen t ly developed for ca lcula- t ing the e lec t ron s t r u c t u r e s of var ious organophosphorus compounds. On the bas i s of the r e su l t s so ob- ta ined we shal l i n t e r p r e t ce r t a in phys ica l and chemica l p r o p e r t i e s of these compounds. Since the ma jo r i t y of o rganophosphorus compounds have a nonplanar s t r uc tu r e , computing methods based on the ~ -approx i - mar ion a r e inappl icable . We used a s e m i e m p i r i c a l method with due al lowance for all the va lence e lec t rons , namely , a modif ied Hoffman i t e ra t ion method incorpora t ing the s e l f - cons i s t ency condition.

A sui table choice of p a r a m e t e r s f o r m s the bas i s of any s e m i e m p i r i c a l method of calculat ing the e lec- t ron s t r u c t u r e of molecu les . All exis t ing s e m i e m p i r i c a l methods a r e cons t ruc ted in such a way that the choice of p a r a m e t e r s is based e i ther on a spec i f ic p rob lem or on the p r o p e r t i e s of a l imi ted number of compounds . An approach of this kind to the solut ion of quan tum-chemica l p r o b l e m s has a number of s e r - ious d i sadvantages . We cons ider the ma in drawbacks to be the following:

a) in solving each p r o b l e m the choice of p a r a m e t e r s nea r l y a lways has to be s t a r t ed a f resh ;

b) a c o m p a r i s o n between the r e su l t s obtained by dif ferent authors is often imposs ib le ;

c) the p r e m i s e s employed as a ba s i s fo r the choice of p a r a m e t e r s a r e often of a spec ia l i zed or even con t rad ic to ry c h a r a c t e r , and this impedes opt imizat ion of the computing technique;

d) the absence of a p r o p e r logical bas i s for the choice of p a r a m e t e r s makes the range of appl icat ion of any p a r t i c u l a r computing method v e r y uncer ta in .

We ou r se lves cons ide r that only the accep tance of a spec i f ic s y s t e m for choosing the p a r a m e t e r s will avoid these sho r t comings . This s y s t e m should be based on a sma l l numbe r of the m o s t genera l p r i n - c ip les . The s y s t e m for choosing the p a r a m e t e r s should const i tu te a se t of ru le s , compr i s ing equations re la t ing one p a r a m e t e r to another , all the p a r a m e t e r s to the c h a r a c t e r i s t i c s of the molecule , and so for th . An analog to this type of s y s t e m is provided by the va r ious ru les which exis t for de te rmin ing the sc reen ing constants in calcula t ing orb i ta l exponents (for example , the f r equen t ly -employed Slater ru les ) .

V. D. Kuzr/etsov Siber ian Phys i co -Techn ica l Ins t i tu te Attached to T o m s k State Univers i ty . T r a n s - lated f r o m Izves t iya Vysshikh Uchebnykh Zavedenii , Fizika, No. 2, pp. 104-109, Feb rua ry , 1974. Or i - ginal a r t i c l e submi t ted November 22, 1972.

�9 19 75 Plenum Publishing Corporation, 22 7 West 17th Street, New York, N. Y. 10011. No part o f this publicaHon may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15.00.

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T h e H o f f m a n m e t h o d h e r e c h o s e n i s conven ien t fo r c o n s t r u c t i n g a p a r a m e t e r - s e l e c t i n g s y s t e m , in tha t i t r e q u i r e s t he s p e c i f i c a t i o n of only two s e t s of p a r a m e t e r s : the o r b i t a l exponen t s and the i o n i z a t i o n p o t e n t i a l s of t he a t o m i c o r b i t s .

2 . T h e o r y o f t h e C o m p u t i n g M e t h o d , a n d S y s t e m

f o r C h o o s i n g t h e P a r a m e t e r s

We d e r i v e the equa t ion fo r the Hof fman i t e r a t i o n m e t h o d f r o m the R o o t h a a n equa t ions [1] fo r m o l e - c u l e s wi th a s y s t e m of c l o s e d s h e l l s :

~t" = det : '?i (ri, :i) [. (l)

m

?i (ril = N ~ ci~X, ( r i ) , (2) z . ~ �9

F~.ci, --- ~i ~ S:,~ci., ~ = I. 2 ..... m, (3) , = 1 " ~ - . i

F~.. -- H~, § G: ..... (4)

H~- tx::(l~ - ~ v~ ~ O_ x. . ( i ldv , . (5) ,Y ..I- - I r l A ]

o, = ~ P,, (~.,l,t) V .%b.. 1

(~i~t) = ~{" x; (i) x (i) • x: (9) x,(2)a-<d~,~ (7) J J / ' 1 2

&.,- (x;(1)x,( l )dv, . (8) O C C

P:~. -= 2 E c~:, ci . (9) i - - ' l

The o r d i n a r y I t o f fman me thod u s e s the fo l lowing p a r a m e t r i z a t i o n [2]:

Hef t = Y;,, (10)

/_/e!f = ! K (Heff ,-~ - He. if ) .%._ (11) ' 2 "

i . e . , the m a t r i x d i a g o n a l e l e m e n t s of the e f f e c t i v e e n e r g y o p e r a t o r H ef t a r e s e t equal to the i o n i z a t i o n p o - t e n t i a l s of the c o r r e s p o n d i n g a t o m i c o r b i t s X~, t a k e n with the o p p o s i t e s ign . The e l e m e n t s I l ~ ff then b e - c o m e i n d e p e n d e n t of the o r b i t a l c o e f f i c i e n t s c i ~ and t h e r e i s no need to i n c o r p o r a t e a s e l f - c o n s i s t e n c y p r o c e d u r e . On the o t h e r hand , th i s k ind of p a r a m e t r i z a t i o n m a k e s no a l l o w a n c e fo r the changes t ak ing p I a c e in the i o n i z a t i o n p o t e n t i a l of t he a t o m i c o r b i t when the a t o m i s i n c o r p o r a t e d into a m o l e c u l e , which g r e a t l y r e d u c e s the r e l i a b i l i t y of the c o m p u t i n g r e s u l t s . R e c e n t y e a r s h a v e s e e n the a p p e a r a n c e of a n u m - b e r of i t e r a t i v e Hof fman m e t h o d s [3, 4] whieh have a l l o w e d fo r t h e m a n n e r in which the i o n i z a t i o n p o t e n - r i a l s of the a t o m i c o r b i t s depend on the p o p u l a t i o n of t he o r b i t s and the c h a r g e on the a t o m , and a l s o fo r the i n f l uence of the p o t e n t i a l s of n e i g h b o r i n g a t o m s . We m a y show tha t the equa t ion fo r Fp4u is d i v i s i b l e in to a s e r i e s of t e r m s r e f l e c t i n g the i nd iv idu a l f a c t o r s :

. . . . , - - v ~ . . . .

A ,

I z ~ = < x : , " 7~-v~ . . . .

L r i . ~

LT, x:, ,. p , , <,~,,~ st> (,.~.~n,,,.o , . t ~ l J

i , z

2 ' ' : ," , , . . _ J

(12)

H e r e the o r b i t Xtz l i e s in t he a t o m A and so do the o r b i t s XX and Xa; t he r e s t be long to o t h e r a t o m s B . It i s w e l l known tha t t h e i o n i z a t i o n p o t e n t i a l fo r a s p e c i f i e d o r b i t in a f r e e a t o m d e p e n d s on a n u m b e r of f a c t o r s , e s p e c i a l l y the p o p u l a t i o n of t he o r b i t in q u e s t i o n and the t o t a l c h a r g e on t h e a t o m . It i s n a t u r a l to a s s u m e s i m i l a r r e l a t i o n s h i p s fo r an a t o m in a m o l e c u l e . In Eq. (12) t he f i r s t two t e r m s r e p r e s e n t the s i n g l e - c e n t e r e d i n t e r a c t i o n s in a t o m A. F o r t h e s e we p r o p o s e the fo l lowing p a r a m e t r i z a t i o n :

235

where

<x:, T v , ......

= --J,,,(1)-~- (,V, .... 1)A,-4- Q,4&,, (13)

Q,~ = X A' : , - zA, (14) t ~ A

J~(1) is the ionizat ion potent ia l of t h e a tomic orbi t X~ with an orbi ta l populat ion of l e ; A# and B bt a r e p a r a m e t e r s which have to be chosen. In Eq. (13) the second t e r m allows for the influence of the popula- t ion of the a tomic orbi t Xt~ on J g . The thi rd t e r m al lows for the change taking p lace in J g with any change in the net cha rge on the a tom A.

The las t two t e r m s in (12) co r r e spond to in te rac t ions with o ther a toms . We a s s u m e d these to be p ropor t iona l to the pa r t p layed by the a tomic orbi t X/~ in fo rming the bonds in the molecu le

< B ~ z ' ~ x , ' - _ ~ P ~ , : .... ( ' , ~ : ' - i : ~ " ) . . . . . _ y ( ' ~ l ' ~ , ' , ) : : a ~ p , , (15)

where

p,~ = ~_~ P~,, S ..... (16) �9 , . :J.

The phys ica l s e n s e of this approx imat ion l ies in the fact that , when the a tomic orbi t X/~ takes pa r t in the bonding, the ionizat ion potent ia l J ~ should a l t e r , s ince the e lec t ron will then take p a r t in i n t e r ac - t ions with o ther a tomic c e n t e r s . The coeff icient 5 A depends on the na ture of the a tom.

Thus we have the following exp re s s ion for F~tt

F..:~ = -- J:~ (1) _ (X:, -- 1) A,~ + QAB.~ _ ~AP:,. (17)

The e x p r e s s i o n for F~v r e m a i n s as be fore , i . e . , Eq. (11). The coeff icient K in (12) is given [5] by

the equat ion

,k'= 2 -. I S , . 1. (18)

The orb i ta l exponents for the a tomic orb i t s X~ w e r e de t e rmined f r o m the Burns [6] ru le s .

Calculat ion of the e l ec t ron s t r u c t u r e was continued until s e l f - cons i s t ency was achieved. At each s tep of the i t e r a t ive p r o c e s s the values of the orb i ta l exponents and m a t r i x e lements Fgv were calculated f r o m the computing data de r ived in the p rev ious i te ra t ion . In o r d e r to improve the convergence , it is e s sen t i a l not to allow sha rp osci l la t ions in the values of F~v; hence the input values for each s u c c e s s i v e s tep in the i t e ra t ion were de t e rmined f r o m the fo rmula

( input),, = (input,),_, --), [( l n p u t ) n - t - - ( output),-t[. (19)

For the orb i ta l exponents the p a r a m e t e r ~ was taken as equal to 0.5; p r a c t i c e in executing the ca l - culat ions showed that this ensured convergence to an a c c u r a c y of no lower than 0.005 a f t e r 5-8 i t e ra t ions . The convergence of Fg~z was cons iderab ly l e s s favorab le and depended on the initial values to a g r e a t e r extent . Fo r Ftz ~ the p a r a m e t e r X was de t e rmined f r o m the equation

k = 1 (20) 3,5 + 0,85 i (input),,_~ -- ( output },J-~ I '

when I ( tnput)n- t - - (output)n-ll <- 2 eV and X = 0.1 fo r g r e a t e r devia t ions .

In calcula t ions ba sed on s e m i e m p i r i e a l methods in which the p a r a m e t e r s depend on the cha rges , it is v e r y i m p o r t a n t t o m a k e a c o r r e c t de te rmina t ion of the cha rges in the orb i t s and a toms . This p r o b l e m has now been cons idered by a n u m b e r of au thors [7, 8], who have shown that the shor tcomings of the Mul- l iken equation [9] for the cha rge in the a tomic orbi t Xtt

OCC m

= X , , , , _:_ 1,

where N = 1 or 2 is the populat ion of the i - th mo lecu l a r orbit , a r e l a rge ly a s soc ia t ed with the fact that in (21) a = 0.5. This co r r e sponds to a s y m m e t r i c a l spl i t t ing of the over lap populat ion between the two a toms ,

236

even if these a toms a r e d i f ferent . In our opinion, the "charge" on the a tom should be taken to m e a n that p a r t of the e lec t ron charge of the molecu le which en ters into the " sphe re of action" of the p a r t i c u l a r a tom. A c r i t e r i on for de te rmin ing such a " sphe re of action" is the orbi ta l e lec t ronegat iv i ty , which c h a r a c t e r i z e s the abi l i ty of the a tom to accept e lec t rons into the a tomic orbi t Xy [10]. On this bas i s we should spl i t the two-orb i t a l t e r m s in (21) in p ropor t ion to the e lec t ronegat iv i t ies of these orb i t s . However , this is p r e - vented by the absence of suff icient ly re l i ab le data as to the e lec t ronega t iv i t i es of the orbi ts c h a r a c t e r i s t i c - a l ly vacant in the f r e e a tom, ' and a lso in negat ive ions. We t h e r e f o r e p ropose spl i t t ing the t e r m s in p r o - por t ion to the ionizat ion potent ia ls of the orb i t s

. . . . . (22) F:,:, + F,.,

The cha rge on the a tom A was de te rmined by Eq. (14).

The ionizat ion potent ia l of an e lec t ron f r o m the i - th m o l e c u l a r orbi t sa t i s f i e s the Coopmans t h e o r e m

Ji = -- ~i. (23)

The energy of an e lec t ron t r ans i t ion f r o m the i - th level to the j - th level is given by the equation

A E i j = ~1 - - :-t �9 (24)

3. C h o i c e o f P a r a m e t e r s . C a l c u l a t i o n s o f

S m a l l M o l e c u l e s

Thus in the method of calcula t ing the e lec t ron s t r u c t u r e s of molecu les h e r e p roposed we have to choose four p a r a m e t e r s for each a tomic orb i t : J y (1), A~, B y and 5 A.

It would at f i r s t appea r that for J y (1), A y and B y the values obtained by Cusachs et al . in [11, 12, 13] would be sui table . However , in [11-13] the values of J y (1), Ay and B y were obtained by calcula t ions re la t ing to f r ee a toms ; fo r a toms in molecu les they may well be d i f ferent . Secondly, in calculat ing the population of the a tomic orb i t s the total population of the a tomic orbi ts is made up of the net population of the a tomic orb i t s and a c e r t a i n p ropor t ion of over lap populat ions, as indicated by Eq. (21). In the case of a f r ee a tom the total populat ion coincides with the net population of the a tomic orbit . The s a m e may be sa id of the cha rge on the a tom.

In o rde r to d e t e r m i n e the p a r a m e t e r s we t h e r e f o r e used a g roup- theo ry p rocedure . The se lec t ion of p a r a m e t e r s was c a r r i e d out for smal l molecu les . These molecu les have a speci f ic s y m m e t r y , and usual ly a cons ide rab le amount of exper imenta l data . Davtyan [14] proposed a g roup- theo ry method en- abling the s y s t e m of m o l e c u l a r orbi ts of a molecule to be s y m m e t r i z e d . For the s y m m e t r i z e d s y s t e m of mo lecu la r orb i t s the m a t r i x of the H a r t r e e - - F o c k ope ra to r was reduced to quasidiagonal fo rm by the m a t r i x - e l e m e n t se lec t ion ru le . The individual blocks of this m a t r i x a r e usual ly of no g r e a t e r than the third o rde r , and on solving these equations with the aid of Eq. (11) we may obtain the re la t ionship between the cor responding orb i ta l energ ies ~i and spec i f i c diagonal m a t r i x e lements Fyy . F u r t h e r m o r e , by using the exper imen ta l values of the orbi ta l energ ies we obtain the des i r ed values of Fyy . A knowledge of the

TABLE 1. Values of the P a r a m e t e r s (eV)

Atom

F

0

N

C

II

Atomic i ' J~(l) A orbit

2p 19,6 3,0 13.8 - ' -

2-i 3,0 2p 17,5 2,3 ,1,3

2p 15,5 , 1,0 10,0 1,50

2s 21,2 I 1,7 7,4 1 2p 11,4 1,5 9,2 ,33

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TABLE 2. Calculated and Exper imenta l Ionization Po-- tent ia ls of the Molecules (eV)

" t Molecule Jcalc (this Jcalc [15] Jexpt [161

paper)

CH, C.~H~ C2H~ [ICN H20 CO CO2 H.~CO

13.20 11,12 11,10 12,74 11,08 13,01 12,8 9,70

13,70 12,24 12,44 13,94 12,88 14,43 15,11 13.36

12,99 10,52 11,40 13,86 12,59 14,01 13,79 10,88

quanti t ies F/~/~ enables us to e s t ima te the values of J/~ (1), A/z, B/~ and 6A. We need then only choose molecu les such that s o m e of the co r r ec t i ons in F~# should be equal to ze ro . For example , in d ia tomic homonuc lea r molecu les the net cha rge of the two a toms equals ze ro and the dependence of Fg~ on B# van i shes . In molecu les of the type XH 2 the mo lecu l a r orbi t of s y m m e t r y b i is one of the np orbi ts of the a tom X, and P c o r r = 0.

Calculat ions re la t ing to s o m e twenty molecules containing hydrogen and e lements of the second per iod enabled us to choose p a r a m e t e r s for the s and p orbi ts of these a toms . The values of the se lec ted p a r a m e t e r s a r e given in Tab le 1. Apar t f rom the p a r a m e t e r s B/~, the values se lec ted by the method h e r e desc r ibed a r e s i m i l a r to the analogous values obtain in [11, 12, 13].

Tab le 2 gives the expe r imen ta l and calcula ted values of the ionization potent ia ls of s e v e r a l m o l e - cu les . The th i rd column contains s o m e r e su l t s taken f r o m [15], in which the calculat ions were c a r r i e d out by a m o r e compl ica ted method.

L I T E R A T U R E C I T E D

1. C . C . J . Roothaan, Rev. Mod. Phys . , 23, 1 (1951). 2. R. Hoffman, J . Chem. Phys . , 39, 1397 (1963). 3. D . G . Ca r ro l l , A. T. A r m s t r o n g , and S. P. McGlinn, J. Chem. Phys . , 44, 1865 (1966). 4. K . C . Wheelock, H. B. Jonassen , and L. C. Cusachs , In ter . J . Quantum Chem. Symp., No. 4, 209

(1971). 5. L . C . Cusaehs , J . Chem. Phys . , 43, 5157 (1965). 6. G. Burns , J . Chem. Phys . , 41, 1521 (1964). 7. I . H . Hi l l i e r and J . F. Whyatt, Inter . J . Quantum Chem., 3, 67 (1969). 8. P. Po l i t z e r and R. R. H a r r i s , J . A m e r . Chem. Soc., 92, 6451 (1970). 9. R . S . Mulliken, J . C h e m . Phys . , 23 , 1833 (1955).

10. J . Hinze and H. H. Jaffe , J . A m e r . Chem. Soc., 84, 540 (1962). 11. L . C . Cusachs and J . W. Reynolds , J . Chem. P h y s . , 43, s160 (1965). 12. L . C . Cusachs , J . W. Reynolds , and D. Ba rna rd , J . Chem. Phys. , 44, 835 (1966). 13. L . C . Cusachs and J . R. Linn, J . Chem. Phys . , 46, 2929 (1967). 14. O . K . Davtyan, Quantum C h e m i s t r y [in Russian] , Izd. Vysshaya Shkola, Moscow (1962). 15. T. Yonezawa, K. Yomaquchi , and H. Kato, Bull. Chem. Soc. Japan, 40, 536 (1967). 16. V . I . Vedeneev, L. V. Gurvich, V. N. Kondra t ' ev , V. A. Medvedev, and L. E. Frankevich ,

Rupture Energy of Chemical Bonds, Ionization Potent ia ls and Elec t ron Affinit ies [in Russian] , Izd. AN SSSR, Moscow (1962).

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