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Excitation energies of Li2-molecule
R a h m a n , A. 1954
Physica XX 623-632
EXCITATION
ENERGIES
OF Li,-MOLECULE
A. RAHMAN *) Institut de physique de l'Universit6, Louvain, Belgique
Synopsis Li z is t r e a t e d as a t w o e l e c t r o n p r o b l e m w i t h t h e a t o m i c K-shells m e r g e d into t h e nuclei. T h e m o l e c u l a r orbitals are b u i l t o u t of 2 s , 2 p x ' v. z a t o m i c orbitals c e n t r e d on each nucleus, t h e e x p o n e n t in t h e 2s and 2p f u n c t i o n s being t h e same. The e x c i t a t i o n energies for t h e t r a n s i t i o n s 1Z+ - - 1~.+, 1/7u __ I:E+ as c a l c u l a t e d b y t h e usual a n t i - s y m m e t r i z e d m o l e c u l a r orbitals, linear c o m b i n a t i o n of a t o m i c orbitals (A.S.M.O~, L.C.A.O.) t h e o r y are g i v e n first and c o m p a r e d w i t h e x p e r i m e n t a l v a l u e s ; as e x p e c t e d t h e r e is wide d i s c r e p a n c y , t h e position being worse at infinite i n t e r n u c l e a r s e p a r a t i o n t h a n in t h e n e i g h b o u r h o o d of t h e e q u i l i b r i u m distance. M o f f i t t ' s i d e a of " A t o m s in M o l e c u l e s " is applied t h e n to this m o d e l for Li v I t is f o u n d t h a t on this simplified m o d e l t h e a p p l i c a t i o n of t h e m e t h o d leads to q u i t e w r o n g v a l u e s for t h e e x c i t a t i o n energies w h e n all t h e c o n f i g u r a t i o n s used in t h e A.S.M.O., L.C.A.O. c a l c u l a t i o n are i n c l u d e d in t h e scheme. I t is f o u n d t h a t on l e a v i n g o u t t h e p u r e l y ionic configurations, i.e., t h e c o n f i g u r a t i o n s r e p r e s e n t i n g Li+ and L i - as i n t e r a c t i n g units, t h e results are as s a t i s f a c t o r y as could be desired in t h e c i r c u m stances, t h e d i s c r e p a n c y b e t w e e n c a l c u l a t e d a n d e x p e r i m e n t a l v a l u e s being always less t h a n 0.3 e.v. A n u n o b s e r v e d h i g h e r 1/7,, s t a t e is i n v e s t i g a t e d s h o w i n g a m i n i m u m of energy.
1. Introduction. To calculate the excitation energies of molecules Moff i t t 1) 2) has shown, with 02 as example, that the most active cause of the discrepancy between calculated and experimental values is that the wave functions used in the calculation lead to extremely bad values of the energies of the respective dissociation products. To get over the difficulty he has shown how to introduce the correct values of the energies of the dissociation products into the usual secular equation, the necessary energy values being taken from experimental data supplied by atomic spectroscopy. Thus in the secular equation modifie'd according to Moffitt, the inaccurate atomic functions are used only for the calculation of the interaction integrals between the atoms when their distance is finite, so that at infinite separation, when these interaction integrals all reduce to zero, the determinant gives the correct values of the purely atomic energies. Applying his method to Oz Moffitt has obtained satisfactory results. F u m i and P a r r s) working on similar lines have shown further the *) On leave from Osmanian University, Hyderabad Decean. India.
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A. RAHMAN
possibility of neglecting certain overlap and interaction integrals without affecting the quality of the calculated results. The purpose here is to apply Moffitt's idea to the Li 2 molecule, treating the problem not in the full complexity of six electrons but simplifying it to a two electron problem with the atomic K-shells merged into the nuclei. As M o f f i t 1) has done for O~, first the results of the usual configurational interaction theory are given to show how these can be improved in following the new method. 2. Li 2 as a two-electron molecule. The main assumption made here is the complete neglect of the K-shells of the two Li atoms. On the one hand, as J a m e s 4) has shown, the presence of these shells is essential for an accurate calculation of each electronic state; on the other hand, the large equilibrium internuclear distance, the low dissociation energy and the depth of the X-ray levels encourages one to regard the effect of the K-shells as essentially additive for the different molecular states; this is specially so when one is interested only in calculating the excitation energies and not each state b y itself. Thus the formation of Liz is considered here as due to two electrons in the presence of two nuclei each of charge unity, and the molecular wave functions are constructed out of 2s, 2p~, 2pv, 2pz atomic functions centred on each nucleus. It is essential to remark the neglect of higher atomic orbitals. The atomic level arising out of a 3s atomic orbital is about as high above the level arising out of a 2p orbital as this latter is above the ground level (2s orbital). As yet there are no tables available for molecular integrals arising out of 3-quantum atomic orbitals, this being the reason for leaving them completely out of account. The choice of a common exponent in the 2s and 2p functions is necessary due to the same reason; this is discussed again in section 4. The tables published b y K o p i n e c k 5) give all the integrals over a satisfactory range of values of the internuclear separation. 3. E x p e r i m e n t a l data on L i v The table in Herzberg's "Diatomic Mole-
cules" (1950) has been used to calculate the Morse parameters for the three electronic states of Liz, 1Y~fl, + 1Xu+ and 1H u. This data is not of a remarkable degree of accuracy, only the few lower vibrational levels being available for the necessary extrapolation; for example the dissociation energy for 11g~ + obtained from molecular beam data is 1.03 e.v. while that obtained from vibrational analysis is 1.12 e.v. This difference according to some authors indicates a van der Waals hump in the 1H u state ; Herzberg has given the necessary references. Using the values quoted in Herzberg's table, the Morse curve, T~ + De { 1 - e-~(r-rel) 2 is given in table 1; in this table,
E X C I T A T I O N E N E R G I E S OF L i ~ - M O L E C U L E
625
le.v. = 8068.3 cm -1, r is in units of a~ = 0.5292 A, the zero of energy being the equilibrium point of zEg. + TABLE I
1//. iS*
Te
De
eli1-1
Cn1-1
20439 14068 0
2955 9326 8486
~e aO 5.5488 5.8711 5.0501
0.5989 0.3193 0.4605
Dissociation
products Li(2P) + Li(2S) Li('P) + l.i(sS) Li(2S) + Li(2S)
The excitation energies calculated from the above figures are illustrated *) in Fig. 68. On making use of the alternative value of D o for zy+ viz. 1.12 e.v., on the scale of Fig. 68 the curves rest unchanged and hence the uncertainties of experimental data mentioned above are, for this calculation, irrelevant. m
Fig. 3. 4. Atomic orbitals. The molecular orbitals were constructed out of the following atomic functions a, = ~/65/3z~ ra exp (-- 6ra)
bs -- x/65/3z~ r b exp (-- 6rb)
ao = ~/~5/~ ra cos 0~ exp (-- ~r~)
b~ = ~/~5/z~r b cos 0b exp (-- &b)
a± = ~/~5/2z~ r, sin 0a exp (--~r~±i~o) b _~= ~/~5/2~ r b sin 0 b exp --~rb-¢-ig) Notice the same exponent 6 in the 2s and 2p functions. The best Slater exponent for the 2s function is 0.66 and for the 2p function it is 0.55; the choice of the same 6, necessary because of the tables available, will be the principal cause of discrepancy with experiment in the calculation of the energies of the dissociation products, and hence, according to Moffitt, also for all values of the internuclear distance. The use of 6 = 0.66 in the 2p functions as in the 2s orbital exaggerates by more t h a n a hundred per cent the difference of energy L i ( 2 p ) - Li(2S). The actual figures appear in the following section. 5. Results o~ A . S . M . O . theory with con/iguration interaction. It is easy to build up anti-symmetrized molecular orbitals (A.S.M.O.) as linear combinations of the atomic orbitals (L.C.A.O.) given in section 4 and it is not necessary to state the procedure in detail. For the states 1E~, + 1E~+ and 1//,, respectively there are 8,5 and 4 interacting configurations. For example, *) Herzberg, loc. cir. p. 147. Physica XX
40
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A. RAHMAN
[a,(1) + b,(1)] [as(2 ) + b,(2)] [a(1) fl(2) --fl(1) a(2)] i an unnormalized configuration of the 1Zo+ family, 1 and 2 referring of :ourse to the two electrons, a and fl being the usual spin functions. Using tables compiled by this author of integra which are all expressible as simple linear combinations of the ir .'grals tabulated by Kopineck, P. C o 1 m a n 6) has made the neces, ;y calculations with 6 = 0.66. The results of these calculations are re7 ~duced in Fig. 1. At
~y_~
0.40
I I
0.30
I
I
0.20
oi" 0.'10 3
"'"-... 4
a,.y_~/? On
5
5
7
8
Fig. 1. Excitation energies 1E+ - - leg+, l//u - - z~+. experimental. A.S.M.O. calculation. infinite separation instead of the experimental value 0.136 R y d (13.6025 eV = 1 Ryd) of the difference of energy {Li(2S) + Li ( 2 p ) } _ {Li(~S) + + Li(2S)}, the value obtained is 0.291 ; however, considering the simple two electron model employed for the calculation, the similarity in shape of the experimental and calculated curves at small distances is remarkable, not forgetting that the comparison here is between curves which arise out of the difference of nearly equal quantities.
627
EXCITATION ENERGIES OF Li2-MOLECULE
Perhaps it is possible to get somewhat better results b y changing the value of the exponent 6 but such calculations have not been made. One might even use a 6 varying with the internuclear distance b u t it is doubtful whether with the simplified model for the molecule such detailed calculations can be of much significance.
6. "Atoms in molecules". The case of Li 2 seems to be particularly well suited for an application of Moffitt's idea; the large equilibrium internuclear separation and the shallow Morse curves indicate that what Moffitt calls the domain of 'valence-coupling' goes down to about 5 a . This is further revealed in Fig. 1 in the manner in which the curve calculated for 1Z~+ - 1~o+ starts to diverge away from the experimental curve. The usual A.S.M.O.L.C.A.O. calculations have shown the nature of the defects arising out of the use of approximate atomic functions and this is what the new method tries to put right. To apply this method it is essential to know from atomic spectroscopy the energy values of the dissociation products which each configuration entails at infinite separation. For example the function mentioned in section 5 viz., (as + b,)(as + bs) can be written as asa8 + asb~ + b~as + b~b~ and at infinite separation describes the following possibilities . . . I) asa, means both electrons on nucleus a, (II) asb~ means electron 1 on a and 2 on b, and SO o n .
Consider for the m o m e n t all possible cases where the two electrons are with different nuclei. Using the data given v) in 'Atomic Energie Levels' (1949) one gets the values given in Table II. TABLE II Diss. products Li(2S) + Li(sS) Li(zS) + Li(2P) Li(sP) + Li(sP)
] I I
Energy (e.v.) --406.878 --405.030 --403.182
Energy (Ryd) --29.9120 --29.7761 --29.6403
Symbol in text
E88 Esa Eea
When the two electrons are on the same nucleus there are similary several possibilities. The wave functions asas, a~ao, asa + etc., illustrate some possibilities; let the energy in each case be denoted b y Iss, Iso, Is+ respectively. Is, for example is the energy of two electrons attached to one of the nuclei, the character of their state being described b y asas, while the other nucleus is bare and at infinite distance from the first. Since here it is a two electron description of Li v I~s is the energy of the dissociation products Li + + Li(both electrons in state 2s). The energy of Li+ is readily available viz., - - 198.049 e.v. The difficulty arises out of Li-, in other words the value of the electron affinity of Li atom and what is more a knowledge of the affinity spectrum is required to get the different I~j. Following a remark of Herzberg
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A. RAHMAN
the electron affinity was put equal to zero and a fortiori its spectrum was overlooked; i.e. the energy of Li- was taken equal to that of Li in its ground state whatever the state of the two electrons in Li-. This gives for all I~, ~198.049 - - 203.439 = 29.5158 × 13.6025 e.v. The initial idea was to use a small variation in the above value of all the I , as an additional parameter; it will turn out later how and why this could not be done. It remains to mention once more t h e exponent 6 in the 2s and 2p functions used in the calculation. As already mentioned this equality of the exponent for 2s and 2p functions is the principal defect in the atomic functions used leading to large discrepancy from experiment for large internuclear distances (see Fig. 1). The purpose of making this calculation is to test the applicability of Moffitt's idea which is precisely that bad atomic functions used only for the calculation of inter-atomic integrals give satisfactory results provided t h e y are not used for calculating the purely intra-atomic integrals, these latter being replaced instead by the experimental data of atomic spectroscopy. One can however reserve the actual value of this exponent as a parameter so as to study whether or not its variation gives better results. 7. M a t r i x e l e m e n t s / o r ' a t o m s i n m o l e c u l e s ' . To illustrate the method it is sufficient to give one example showing how the matrix elements are calculated in following Moffitt. Two of the eight configurations of the state 1 Za+ are, without the anti-symmetric spin factor, ~1 = (as + bs) (as + bs) and Tg. -~ (as + b s) (ao + bo) + (ao + b~) (as + b s). The Hamiltonian of the simplified problem is H = --
V~ - -
V~ +
( 2 / r l , - - 2 / r . 1 - - 2/r,~. ~
2/rbl --
unit of energy --13.6025 e.v. Following Moffitt the result of operating by H on ~ respectively as
2/rb,),
and ~
is written
Iss(asas + bsbs) + Ess(asbs + bsas) --2/rax(bsb s + bsa~) --2/raz(bsb s + asbs) = --2/rbx(asa s + asbs) --2/rb2(asa s + bsa,) + 2/rx,(asbs + bsas) and
Isa(asaa + bsba + --2/ral(bsb~ + --2/rbl(asaa + + 2/rx2(asba +
aaas + bobs) + Es.(asba + bsa. + aab, + baas) babs + bsaa+ baas)--2/ra~(bsba + bab8 + asba + aabs) aaas + asba + aabs)--2/rb2(asaa + aaas + bsaa + baas) aabs + bsaa + baas)
To construct the matrix elements f ~1H~1, f ~ , H ~ I , f ~ I H ~ , , f ~ ' H ~ it is simply necessary to substitute for H ~ z and HkV2 the above two expressions and to use ½(f ~ 2 H T 1 + f ~ I H T , ) t h u s c a l c u l a t e d as the (1, 2)
EXCITATION ENERGIES OF Li2-MOLECULE
629
element in the (symmetric) secular determinant for the energy. In thfs manner all elements can be written down as q u i t e simple expressions involving integrals tabulated by Kopineck and no secondary tabulation is necessary. The number of these expressions being quite large they are not reproduced here.
8. The negative result o/ the calculation and its remedy. The secular determinants thus formulated on being solved for the lowest.root with a value of ~ = 0.7 gave absurdly bad results for the excitation energies. The Ryd
0.40
_
l
o.ao
--
\
t
I
\ I
a~yrnp
0.20
_
\
" ' ' - - .
".~-.. 0.10
3 F i g . 2.
_
4
~,Y~3
I
I '",
I
-'I ......
4
5
6
7
t *t'~
Qe
8
Excitation energies I Z + - - lZ+, 1/-/u - - IZ+. experimental. 'atoms in molecules' calculation (covalent). 'atoms in molecules' calculation (covalent and ionic).
value of ~ chosen had no special significance since the final intention was to vary it to get the best results. Anyhow it was thought best to start off with a value of ~ in the neighbourhood of the best Slater orbital value of 0.66 for the ~S state of Li. Whereas the original hope was to get an improvement
630
A. RAHMAN
on the curves presented in Fig. 1, the calculations gave results quite to the contrary. With ~ = 0.7, at R = 5a 0, the difference 1Z~+ - - 1E~+ turned out to be so large t h a t on the scale of Fig. 1 the point will fall far outside the figure. The only parameter available was 6 (since I,j, as already pointed out, could not be freely varied), and its variation showed t h a t at R = 5 a o for about 1.33 it was possible to get the calculated differences 1Z+u - - ~Zg+ and 1 H u - - 1Zg+ near the experimental value. But this proved to be a rather artificial sort of improvement because when the calculation was made for a range of values of R, with 6---- 1.33, the gradient of the calculated and experimental curves turned out to be absolutely different in both cases; the results are shown in Fig. 2 (dotted curve). It was obvious at this stage t h a t there was a disturbing element in the calculation which had to be located before any conclusions could be arrived at regarding the method and the problem itself. This Was achieved in changing over to pure ionic and pure covalent functions from the ones used above. Since this only means taking a certain linear combination of the functions previously used the energy values calculated from each secular determinant remain the same as before. For example (as + bs) (a e + be) and (as - - bs) (ae - - be) are two orbitals of the l y + family. On taking their sum and difference one gets, apart from factor 2, (aea e + aeae) and (aebe + bsas) these new functions being obviously pure ionic and covalent orbitals as the former were mixtures of the two. When the same is done for the whole set of functions used in each family one gets for ~Zu, + 4 covalent and 4 ionic configurations, for ~]~u, + 1 and 4 respectively and for 1/~u, 2 and 2 respectively. In each case one of the covalent configurations gives the respective dissociation products; these configurations are for for for
1X+, aeb e + bsa e giving dissociation products Li(2S)+Li(2S) 1E~,, + aebo--beao--aabe+baa . . . . . . . Li(2S) +Li(~P) 1H~,, aeb÷+b~a++a+bs+b+ae . . . . . . Li(2S)+Li(ap)
Hence for sufficiently large internuclear distances, the above three configurations will automatically be the most dominant as far as the lowest root of each secular determinant is concerned; the presence of the other configurations will change but slightly (in pushing it downwards) the value obtained from the dominant configuration alone. In the case of the A.S.M.O., L.C.A.O. calculation it was found t h a t certainly down to a distance of 5a~ between the nuclei, the above remain the dominant configurations in each case, the addition of the rest of the configurations producing only a depression of about one per cent. Afortiori in the case of 'Atoms in Molecules' the same must be found to occur. In fact a close investigation of the secular determinants formulated in terms of pure ionic and covalent functions revealed that this was not true. It was expected t h a t in the case of 1E~+ for
EXCITATION ENERGIES OF Li2-MOLECULE
631
example, when each term along the diagonal is taken in isolation, the one arising out of the function (asb, + bsa~) would give the lowest value of the energy, this being lowered to a certain extent when all configurations are taken together due to their interaction. As indicated above, this expectation was verified in. the case of the usual theory. The following figures will illustrate the circumstances prevailing in the case of Moffitt's theory. Three of the eight 1Z+ configurations are
~1 = a~b~ + b~a~ (covalent) a°ao + b~,b~, (ionic) ~3 = a sa~ + aoa, + b~b, + b,b~ (ionic) -_
At R ---- 5 a , with 6 = 0.7, calculation shows that #1 alone gives --30.387 R y d ¢2 alone gives --30.475 R #a alone gives --30.483 R
~bI and ~b2 together give --30.503 R #1 and Cs together give --30.512 R #~ and #3 together give --30.682 R
#~, #~. and ¢8 all together give a value very slightly lower than --30.682, for the energy of ~Zg. + The above figures can be described b y saying that the energy given by the two ionic configurations is slightly lowered b y the inclusion of the higher configuration #~ which is covalent ! A variation in the value of I , necessary to reverse the above unexpected behaviour is so large and positive that firstly b y itself the value is physically inadmissible and secondly such a large upward shift of the ionic levels with the omission of the 3s covalent levels cannot be permitted. It is difficult to see why in using Moffitt's method on the two electron model of the molecule such an unexpected behaviour of the ionic configurations comes about. Further, as is shown in the following section, if these configurations are throughout left out of account and only the covalent configurations are preserved quite satisfactory results are obtained.
9. Only covalent configurations. Once the ionic configurations were left out of account the problem was readily solved. Due to a reduction in the number of configurations the computational work was quite distinctly less. According to the original intention the exponent 6 was used as a variable parameter; starting from 6 = 0.7 it was found that one had to go upto 6 = 1.0 to produce a satisfactory fit with the experimental curves. The results of the calculation with 6 = 1.0 are illustrated in Fig. 2. It is possible that if 6 is varied with the nuclear separation an even better fit can be obtained but in view of the limited number of observed states for comparison such a fit would not be of much significance. As expected, below 5a 0 the curves diverge away from the experimental curves, in this region what Moffitt calls 'valence-coupling' being not valid anymore. Otherwise the divergence from the experimental curves is generally less than about 0.25 electron volts for both curves.
6~2
E X C I T A T I O N E N E R G I E S OF
Liz
MOLECULE
As a matter of further interest Fig. 2 also illustrates the excitation energy to an excited 1//u level, one which dissociates into Li(2p) + Li(2P). Using this calculated excitation energy curve and the experimental energy curve for the state 1E+, the energy curve for this x//~ state exhibits a slight stability the minimum being at about 6.4a. 10. Conclusion. In making use of Moffitt's 'Atoms in Molecules', it seems that the simplifying approximations made in dealing with a molecular problem might entail certain features which are physically inadmissible. The calculations rep.orted here show that considering Li 2 as a two electronic problem and applying Moffitt's idea in a straightforward way leads to physically unacceptable circumstances and hence to wrong results. If care is taken to keep out such circumstances, Moffitt's method leads to satisfactory results. Considering the behaviour of the ionic configurations in the calculations reported here it will be unwise to make any generalisations; for example in the case of 02 treated successfully b y M o f f i t t ~), it is precisely the manipulation of an ionic configuration which leads to satisfactory results and not its exclusion. It is perhaps worthwhile to mention in the light of the above results, that the Heitler-London scheme of treating configuration interaction is more suitable for the application of Moffitt's method than the Hund-Mulliken scheme and this is quite natural, for Moffitt's 'Atoms in Molecules' is in fact. a generalisation of the simple Heitler-London treatment. Further, the behaviour of the ionic configurations in the above calculation shows that for MoffitVs method to work it is not sufficient to have an asymptotically good behaviour; it is at the same time necessary to use wave functions which describe the electronic distribution round each nucleus with a certain minimum degree of accuracy; it is seen above that the wave .functions corresponding to ionic configurations fall below the minimum requirements necessary and produce quite absurd results when used for 'Atoms in Molecules'. More applications of Moffitt's method, will h e l p to clear up the points illustrated here, and perhaps will lead to a quantitative formulation of the "minimum requirements" just mentioned.
Acknowledgements. The author wishes to thank professor C. M a n n eb a c k for many useful discussions during the progress of the above work; the author is also grateful to the "Centre de chimie-physique mol~culaire" for financial support during the authors stay in Belgium. Received 21-6-54. 1) 2) 3) 4) 5) -6) 7)
Moffitt, Moffitt, Fumi, James, Kop in CoIm a National
REFERENCES W., Proc. roy. Soc., A, 210 (1951) 224. W., Proc. roy. Soc., A, ,°I0 (1951) 245. F. G. and P a r r , R. G., J. chem. Phys. 21 (1953) 1864. H., J. chem. Phys. 2 (1934) 794. e c k , H. J., Z. Naturforschg. 5a (1950) 420; 6a (1951) 177; 7a (1952) 785. n, P., Th~se Licence, University of Louvain, 1954. Bureau of Standards. W a s h i n g t o n (1949). Circular No. 467.
Physica XX 623-632
EXCITATION
ENERGIES
OF Li,-MOLECULE
A. RAHMAN *) Institut de physique de l'Universit6, Louvain, Belgique
Synopsis Li z is t r e a t e d as a t w o e l e c t r o n p r o b l e m w i t h t h e a t o m i c K-shells m e r g e d into t h e nuclei. T h e m o l e c u l a r orbitals are b u i l t o u t of 2 s , 2 p x ' v. z a t o m i c orbitals c e n t r e d on each nucleus, t h e e x p o n e n t in t h e 2s and 2p f u n c t i o n s being t h e same. The e x c i t a t i o n energies for t h e t r a n s i t i o n s 1Z+ - - 1~.+, 1/7u __ I:E+ as c a l c u l a t e d b y t h e usual a n t i - s y m m e t r i z e d m o l e c u l a r orbitals, linear c o m b i n a t i o n of a t o m i c orbitals (A.S.M.O~, L.C.A.O.) t h e o r y are g i v e n first and c o m p a r e d w i t h e x p e r i m e n t a l v a l u e s ; as e x p e c t e d t h e r e is wide d i s c r e p a n c y , t h e position being worse at infinite i n t e r n u c l e a r s e p a r a t i o n t h a n in t h e n e i g h b o u r h o o d of t h e e q u i l i b r i u m distance. M o f f i t t ' s i d e a of " A t o m s in M o l e c u l e s " is applied t h e n to this m o d e l for Li v I t is f o u n d t h a t on this simplified m o d e l t h e a p p l i c a t i o n of t h e m e t h o d leads to q u i t e w r o n g v a l u e s for t h e e x c i t a t i o n energies w h e n all t h e c o n f i g u r a t i o n s used in t h e A.S.M.O., L.C.A.O. c a l c u l a t i o n are i n c l u d e d in t h e scheme. I t is f o u n d t h a t on l e a v i n g o u t t h e p u r e l y ionic configurations, i.e., t h e c o n f i g u r a t i o n s r e p r e s e n t i n g Li+ and L i - as i n t e r a c t i n g units, t h e results are as s a t i s f a c t o r y as could be desired in t h e c i r c u m stances, t h e d i s c r e p a n c y b e t w e e n c a l c u l a t e d a n d e x p e r i m e n t a l v a l u e s being always less t h a n 0.3 e.v. A n u n o b s e r v e d h i g h e r 1/7,, s t a t e is i n v e s t i g a t e d s h o w i n g a m i n i m u m of energy.
1. Introduction. To calculate the excitation energies of molecules Moff i t t 1) 2) has shown, with 02 as example, that the most active cause of the discrepancy between calculated and experimental values is that the wave functions used in the calculation lead to extremely bad values of the energies of the respective dissociation products. To get over the difficulty he has shown how to introduce the correct values of the energies of the dissociation products into the usual secular equation, the necessary energy values being taken from experimental data supplied by atomic spectroscopy. Thus in the secular equation modifie'd according to Moffitt, the inaccurate atomic functions are used only for the calculation of the interaction integrals between the atoms when their distance is finite, so that at infinite separation, when these interaction integrals all reduce to zero, the determinant gives the correct values of the purely atomic energies. Applying his method to Oz Moffitt has obtained satisfactory results. F u m i and P a r r s) working on similar lines have shown further the *) On leave from Osmanian University, Hyderabad Decean. India.
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624
A. RAHMAN
possibility of neglecting certain overlap and interaction integrals without affecting the quality of the calculated results. The purpose here is to apply Moffitt's idea to the Li 2 molecule, treating the problem not in the full complexity of six electrons but simplifying it to a two electron problem with the atomic K-shells merged into the nuclei. As M o f f i t 1) has done for O~, first the results of the usual configurational interaction theory are given to show how these can be improved in following the new method. 2. Li 2 as a two-electron molecule. The main assumption made here is the complete neglect of the K-shells of the two Li atoms. On the one hand, as J a m e s 4) has shown, the presence of these shells is essential for an accurate calculation of each electronic state; on the other hand, the large equilibrium internuclear distance, the low dissociation energy and the depth of the X-ray levels encourages one to regard the effect of the K-shells as essentially additive for the different molecular states; this is specially so when one is interested only in calculating the excitation energies and not each state b y itself. Thus the formation of Liz is considered here as due to two electrons in the presence of two nuclei each of charge unity, and the molecular wave functions are constructed out of 2s, 2p~, 2pv, 2pz atomic functions centred on each nucleus. It is essential to remark the neglect of higher atomic orbitals. The atomic level arising out of a 3s atomic orbital is about as high above the level arising out of a 2p orbital as this latter is above the ground level (2s orbital). As yet there are no tables available for molecular integrals arising out of 3-quantum atomic orbitals, this being the reason for leaving them completely out of account. The choice of a common exponent in the 2s and 2p functions is necessary due to the same reason; this is discussed again in section 4. The tables published b y K o p i n e c k 5) give all the integrals over a satisfactory range of values of the internuclear separation. 3. E x p e r i m e n t a l data on L i v The table in Herzberg's "Diatomic Mole-
cules" (1950) has been used to calculate the Morse parameters for the three electronic states of Liz, 1Y~fl, + 1Xu+ and 1H u. This data is not of a remarkable degree of accuracy, only the few lower vibrational levels being available for the necessary extrapolation; for example the dissociation energy for 11g~ + obtained from molecular beam data is 1.03 e.v. while that obtained from vibrational analysis is 1.12 e.v. This difference according to some authors indicates a van der Waals hump in the 1H u state ; Herzberg has given the necessary references. Using the values quoted in Herzberg's table, the Morse curve, T~ + De { 1 - e-~(r-rel) 2 is given in table 1; in this table,
E X C I T A T I O N E N E R G I E S OF L i ~ - M O L E C U L E
625
le.v. = 8068.3 cm -1, r is in units of a~ = 0.5292 A, the zero of energy being the equilibrium point of zEg. + TABLE I
1//. iS*
Te
De
eli1-1
Cn1-1
20439 14068 0
2955 9326 8486
~e aO 5.5488 5.8711 5.0501
0.5989 0.3193 0.4605
Dissociation
products Li(2P) + Li(2S) Li('P) + l.i(sS) Li(2S) + Li(2S)
The excitation energies calculated from the above figures are illustrated *) in Fig. 68. On making use of the alternative value of D o for zy+ viz. 1.12 e.v., on the scale of Fig. 68 the curves rest unchanged and hence the uncertainties of experimental data mentioned above are, for this calculation, irrelevant. m
Fig. 3. 4. Atomic orbitals. The molecular orbitals were constructed out of the following atomic functions a, = ~/65/3z~ ra exp (-- 6ra)
bs -- x/65/3z~ r b exp (-- 6rb)
ao = ~/~5/~ ra cos 0~ exp (-- ~r~)
b~ = ~/~5/z~r b cos 0b exp (-- &b)
a± = ~/~5/2z~ r, sin 0a exp (--~r~±i~o) b _~= ~/~5/2~ r b sin 0 b exp --~rb-¢-ig) Notice the same exponent 6 in the 2s and 2p functions. The best Slater exponent for the 2s function is 0.66 and for the 2p function it is 0.55; the choice of the same 6, necessary because of the tables available, will be the principal cause of discrepancy with experiment in the calculation of the energies of the dissociation products, and hence, according to Moffitt, also for all values of the internuclear distance. The use of 6 = 0.66 in the 2p functions as in the 2s orbital exaggerates by more t h a n a hundred per cent the difference of energy L i ( 2 p ) - Li(2S). The actual figures appear in the following section. 5. Results o~ A . S . M . O . theory with con/iguration interaction. It is easy to build up anti-symmetrized molecular orbitals (A.S.M.O.) as linear combinations of the atomic orbitals (L.C.A.O.) given in section 4 and it is not necessary to state the procedure in detail. For the states 1E~, + 1E~+ and 1//,, respectively there are 8,5 and 4 interacting configurations. For example, *) Herzberg, loc. cir. p. 147. Physica XX
40
626
A. RAHMAN
[a,(1) + b,(1)] [as(2 ) + b,(2)] [a(1) fl(2) --fl(1) a(2)] i an unnormalized configuration of the 1Zo+ family, 1 and 2 referring of :ourse to the two electrons, a and fl being the usual spin functions. Using tables compiled by this author of integra which are all expressible as simple linear combinations of the ir .'grals tabulated by Kopineck, P. C o 1 m a n 6) has made the neces, ;y calculations with 6 = 0.66. The results of these calculations are re7 ~duced in Fig. 1. At
~y_~
0.40
I I
0.30
I
I
0.20
oi" 0.'10 3
"'"-... 4
a,.y_~/? On
5
5
7
8
Fig. 1. Excitation energies 1E+ - - leg+, l//u - - z~+. experimental. A.S.M.O. calculation. infinite separation instead of the experimental value 0.136 R y d (13.6025 eV = 1 Ryd) of the difference of energy {Li(2S) + Li ( 2 p ) } _ {Li(~S) + + Li(2S)}, the value obtained is 0.291 ; however, considering the simple two electron model employed for the calculation, the similarity in shape of the experimental and calculated curves at small distances is remarkable, not forgetting that the comparison here is between curves which arise out of the difference of nearly equal quantities.
627
EXCITATION ENERGIES OF Li2-MOLECULE
Perhaps it is possible to get somewhat better results b y changing the value of the exponent 6 but such calculations have not been made. One might even use a 6 varying with the internuclear distance b u t it is doubtful whether with the simplified model for the molecule such detailed calculations can be of much significance.
6. "Atoms in molecules". The case of Li 2 seems to be particularly well suited for an application of Moffitt's idea; the large equilibrium internuclear separation and the shallow Morse curves indicate that what Moffitt calls the domain of 'valence-coupling' goes down to about 5 a . This is further revealed in Fig. 1 in the manner in which the curve calculated for 1Z~+ - 1~o+ starts to diverge away from the experimental curve. The usual A.S.M.O.L.C.A.O. calculations have shown the nature of the defects arising out of the use of approximate atomic functions and this is what the new method tries to put right. To apply this method it is essential to know from atomic spectroscopy the energy values of the dissociation products which each configuration entails at infinite separation. For example the function mentioned in section 5 viz., (as + b,)(as + bs) can be written as asa8 + asb~ + b~as + b~b~ and at infinite separation describes the following possibilities . . . I) asa, means both electrons on nucleus a, (II) asb~ means electron 1 on a and 2 on b, and SO o n .
Consider for the m o m e n t all possible cases where the two electrons are with different nuclei. Using the data given v) in 'Atomic Energie Levels' (1949) one gets the values given in Table II. TABLE II Diss. products Li(2S) + Li(sS) Li(zS) + Li(2P) Li(sP) + Li(sP)
] I I
Energy (e.v.) --406.878 --405.030 --403.182
Energy (Ryd) --29.9120 --29.7761 --29.6403
Symbol in text
E88 Esa Eea
When the two electrons are on the same nucleus there are similary several possibilities. The wave functions asas, a~ao, asa + etc., illustrate some possibilities; let the energy in each case be denoted b y Iss, Iso, Is+ respectively. Is, for example is the energy of two electrons attached to one of the nuclei, the character of their state being described b y asas, while the other nucleus is bare and at infinite distance from the first. Since here it is a two electron description of Li v I~s is the energy of the dissociation products Li + + Li(both electrons in state 2s). The energy of Li+ is readily available viz., - - 198.049 e.v. The difficulty arises out of Li-, in other words the value of the electron affinity of Li atom and what is more a knowledge of the affinity spectrum is required to get the different I~j. Following a remark of Herzberg
628
A. RAHMAN
the electron affinity was put equal to zero and a fortiori its spectrum was overlooked; i.e. the energy of Li- was taken equal to that of Li in its ground state whatever the state of the two electrons in Li-. This gives for all I~, ~198.049 - - 203.439 = 29.5158 × 13.6025 e.v. The initial idea was to use a small variation in the above value of all the I , as an additional parameter; it will turn out later how and why this could not be done. It remains to mention once more t h e exponent 6 in the 2s and 2p functions used in the calculation. As already mentioned this equality of the exponent for 2s and 2p functions is the principal defect in the atomic functions used leading to large discrepancy from experiment for large internuclear distances (see Fig. 1). The purpose of making this calculation is to test the applicability of Moffitt's idea which is precisely that bad atomic functions used only for the calculation of inter-atomic integrals give satisfactory results provided t h e y are not used for calculating the purely intra-atomic integrals, these latter being replaced instead by the experimental data of atomic spectroscopy. One can however reserve the actual value of this exponent as a parameter so as to study whether or not its variation gives better results. 7. M a t r i x e l e m e n t s / o r ' a t o m s i n m o l e c u l e s ' . To illustrate the method it is sufficient to give one example showing how the matrix elements are calculated in following Moffitt. Two of the eight configurations of the state 1 Za+ are, without the anti-symmetric spin factor, ~1 = (as + bs) (as + bs) and Tg. -~ (as + b s) (ao + bo) + (ao + b~) (as + b s). The Hamiltonian of the simplified problem is H = --
V~ - -
V~ +
( 2 / r l , - - 2 / r . 1 - - 2/r,~. ~
2/rbl --
unit of energy --13.6025 e.v. Following Moffitt the result of operating by H on ~ respectively as
2/rb,),
and ~
is written
Iss(asas + bsbs) + Ess(asbs + bsas) --2/rax(bsb s + bsa~) --2/raz(bsb s + asbs) = --2/rbx(asa s + asbs) --2/rb2(asa s + bsa,) + 2/rx,(asbs + bsas) and
Isa(asaa + bsba + --2/ral(bsb~ + --2/rbl(asaa + + 2/rx2(asba +
aaas + bobs) + Es.(asba + bsa. + aab, + baas) babs + bsaa+ baas)--2/ra~(bsba + bab8 + asba + aabs) aaas + asba + aabs)--2/rb2(asaa + aaas + bsaa + baas) aabs + bsaa + baas)
To construct the matrix elements f ~1H~1, f ~ , H ~ I , f ~ I H ~ , , f ~ ' H ~ it is simply necessary to substitute for H ~ z and HkV2 the above two expressions and to use ½(f ~ 2 H T 1 + f ~ I H T , ) t h u s c a l c u l a t e d as the (1, 2)
EXCITATION ENERGIES OF Li2-MOLECULE
629
element in the (symmetric) secular determinant for the energy. In thfs manner all elements can be written down as q u i t e simple expressions involving integrals tabulated by Kopineck and no secondary tabulation is necessary. The number of these expressions being quite large they are not reproduced here.
8. The negative result o/ the calculation and its remedy. The secular determinants thus formulated on being solved for the lowest.root with a value of ~ = 0.7 gave absurdly bad results for the excitation energies. The Ryd
0.40
_
l
o.ao
--
\
t
I
\ I
a~yrnp
0.20
_
\
" ' ' - - .
".~-.. 0.10
3 F i g . 2.
_
4
~,Y~3
I
I '",
I
-'I ......
4
5
6
7
t *t'~
Qe
8
Excitation energies I Z + - - lZ+, 1/-/u - - IZ+. experimental. 'atoms in molecules' calculation (covalent). 'atoms in molecules' calculation (covalent and ionic).
value of ~ chosen had no special significance since the final intention was to vary it to get the best results. Anyhow it was thought best to start off with a value of ~ in the neighbourhood of the best Slater orbital value of 0.66 for the ~S state of Li. Whereas the original hope was to get an improvement
630
A. RAHMAN
on the curves presented in Fig. 1, the calculations gave results quite to the contrary. With ~ = 0.7, at R = 5a 0, the difference 1Z~+ - - 1E~+ turned out to be so large t h a t on the scale of Fig. 1 the point will fall far outside the figure. The only parameter available was 6 (since I,j, as already pointed out, could not be freely varied), and its variation showed t h a t at R = 5 a o for about 1.33 it was possible to get the calculated differences 1Z+u - - ~Zg+ and 1 H u - - 1Zg+ near the experimental value. But this proved to be a rather artificial sort of improvement because when the calculation was made for a range of values of R, with 6---- 1.33, the gradient of the calculated and experimental curves turned out to be absolutely different in both cases; the results are shown in Fig. 2 (dotted curve). It was obvious at this stage t h a t there was a disturbing element in the calculation which had to be located before any conclusions could be arrived at regarding the method and the problem itself. This Was achieved in changing over to pure ionic and pure covalent functions from the ones used above. Since this only means taking a certain linear combination of the functions previously used the energy values calculated from each secular determinant remain the same as before. For example (as + bs) (a e + be) and (as - - bs) (ae - - be) are two orbitals of the l y + family. On taking their sum and difference one gets, apart from factor 2, (aea e + aeae) and (aebe + bsas) these new functions being obviously pure ionic and covalent orbitals as the former were mixtures of the two. When the same is done for the whole set of functions used in each family one gets for ~Zu, + 4 covalent and 4 ionic configurations, for ~]~u, + 1 and 4 respectively and for 1/~u, 2 and 2 respectively. In each case one of the covalent configurations gives the respective dissociation products; these configurations are for for for
1X+, aeb e + bsa e giving dissociation products Li(2S)+Li(2S) 1E~,, + aebo--beao--aabe+baa . . . . . . . Li(2S) +Li(~P) 1H~,, aeb÷+b~a++a+bs+b+ae . . . . . . Li(2S)+Li(ap)
Hence for sufficiently large internuclear distances, the above three configurations will automatically be the most dominant as far as the lowest root of each secular determinant is concerned; the presence of the other configurations will change but slightly (in pushing it downwards) the value obtained from the dominant configuration alone. In the case of the A.S.M.O., L.C.A.O. calculation it was found t h a t certainly down to a distance of 5a~ between the nuclei, the above remain the dominant configurations in each case, the addition of the rest of the configurations producing only a depression of about one per cent. Afortiori in the case of 'Atoms in Molecules' the same must be found to occur. In fact a close investigation of the secular determinants formulated in terms of pure ionic and covalent functions revealed that this was not true. It was expected t h a t in the case of 1E~+ for
EXCITATION ENERGIES OF Li2-MOLECULE
631
example, when each term along the diagonal is taken in isolation, the one arising out of the function (asb, + bsa~) would give the lowest value of the energy, this being lowered to a certain extent when all configurations are taken together due to their interaction. As indicated above, this expectation was verified in. the case of the usual theory. The following figures will illustrate the circumstances prevailing in the case of Moffitt's theory. Three of the eight 1Z+ configurations are
~1 = a~b~ + b~a~ (covalent) a°ao + b~,b~, (ionic) ~3 = a sa~ + aoa, + b~b, + b,b~ (ionic) -_
At R ---- 5 a , with 6 = 0.7, calculation shows that #1 alone gives --30.387 R y d ¢2 alone gives --30.475 R #a alone gives --30.483 R
~bI and ~b2 together give --30.503 R #1 and Cs together give --30.512 R #~ and #3 together give --30.682 R
#~, #~. and ¢8 all together give a value very slightly lower than --30.682, for the energy of ~Zg. + The above figures can be described b y saying that the energy given by the two ionic configurations is slightly lowered b y the inclusion of the higher configuration #~ which is covalent ! A variation in the value of I , necessary to reverse the above unexpected behaviour is so large and positive that firstly b y itself the value is physically inadmissible and secondly such a large upward shift of the ionic levels with the omission of the 3s covalent levels cannot be permitted. It is difficult to see why in using Moffitt's method on the two electron model of the molecule such an unexpected behaviour of the ionic configurations comes about. Further, as is shown in the following section, if these configurations are throughout left out of account and only the covalent configurations are preserved quite satisfactory results are obtained.
9. Only covalent configurations. Once the ionic configurations were left out of account the problem was readily solved. Due to a reduction in the number of configurations the computational work was quite distinctly less. According to the original intention the exponent 6 was used as a variable parameter; starting from 6 = 0.7 it was found that one had to go upto 6 = 1.0 to produce a satisfactory fit with the experimental curves. The results of the calculation with 6 = 1.0 are illustrated in Fig. 2. It is possible that if 6 is varied with the nuclear separation an even better fit can be obtained but in view of the limited number of observed states for comparison such a fit would not be of much significance. As expected, below 5a 0 the curves diverge away from the experimental curves, in this region what Moffitt calls 'valence-coupling' being not valid anymore. Otherwise the divergence from the experimental curves is generally less than about 0.25 electron volts for both curves.
6~2
E X C I T A T I O N E N E R G I E S OF
Liz
MOLECULE
As a matter of further interest Fig. 2 also illustrates the excitation energy to an excited 1//u level, one which dissociates into Li(2p) + Li(2P). Using this calculated excitation energy curve and the experimental energy curve for the state 1E+, the energy curve for this x//~ state exhibits a slight stability the minimum being at about 6.4a. 10. Conclusion. In making use of Moffitt's 'Atoms in Molecules', it seems that the simplifying approximations made in dealing with a molecular problem might entail certain features which are physically inadmissible. The calculations rep.orted here show that considering Li 2 as a two electronic problem and applying Moffitt's idea in a straightforward way leads to physically unacceptable circumstances and hence to wrong results. If care is taken to keep out such circumstances, Moffitt's method leads to satisfactory results. Considering the behaviour of the ionic configurations in the calculations reported here it will be unwise to make any generalisations; for example in the case of 02 treated successfully b y M o f f i t t ~), it is precisely the manipulation of an ionic configuration which leads to satisfactory results and not its exclusion. It is perhaps worthwhile to mention in the light of the above results, that the Heitler-London scheme of treating configuration interaction is more suitable for the application of Moffitt's method than the Hund-Mulliken scheme and this is quite natural, for Moffitt's 'Atoms in Molecules' is in fact. a generalisation of the simple Heitler-London treatment. Further, the behaviour of the ionic configurations in the above calculation shows that for MoffitVs method to work it is not sufficient to have an asymptotically good behaviour; it is at the same time necessary to use wave functions which describe the electronic distribution round each nucleus with a certain minimum degree of accuracy; it is seen above that the wave .functions corresponding to ionic configurations fall below the minimum requirements necessary and produce quite absurd results when used for 'Atoms in Molecules'. More applications of Moffitt's method, will h e l p to clear up the points illustrated here, and perhaps will lead to a quantitative formulation of the "minimum requirements" just mentioned.
Acknowledgements. The author wishes to thank professor C. M a n n eb a c k for many useful discussions during the progress of the above work; the author is also grateful to the "Centre de chimie-physique mol~culaire" for financial support during the authors stay in Belgium. Received 21-6-54. 1) 2) 3) 4) 5) -6) 7)
Moffitt, Moffitt, Fumi, James, Kop in CoIm a National
REFERENCES W., Proc. roy. Soc., A, 210 (1951) 224. W., Proc. roy. Soc., A, ,°I0 (1951) 245. F. G. and P a r r , R. G., J. chem. Phys. 21 (1953) 1864. H., J. chem. Phys. 2 (1934) 794. e c k , H. J., Z. Naturforschg. 5a (1950) 420; 6a (1951) 177; 7a (1952) 785. n, P., Th~se Licence, University of Louvain, 1954. Bureau of Standards. W a s h i n g t o n (1949). Circular No. 467.