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Finite-field MC SCF study of the hydrogen molecule polarizability
Volume
6T. number
CIfII.\IIC_AL PHYSICS
I
1 Norember
LE-ITERS
1979
FINITE-FIELD MC SCF STUDY OF THE HYDROGEN MOLECULE POLARIZABILITY
Rcceired
20 JoI:. I979
The polxiubilit~ of the II2 molecule IS cornputrd usmg the tinitc-ticld method .md rclattwl) simple multiconfiiuraticn ‘33‘ ~~.tvefttnctions_ .An.d~ sing the drpcndcncc of the polkzabdity on the internuclear dntnnce It is shoxbn that rxithin the chnxn appruGm.!tmn the correhtlon contrtbutions to the poIariz.tbiIit> can be cstimJted niten they become signifktnt.
The finite-field perturbation theory 11~sbeen widely used to study .ttomic and molecular pohrriz~bilities. In particular, It was recently applied to analyse the correlation corrections to the static dipole polarizsbilities of small molecules [ I--3] _ We refer to these papers for d thorough discussion of the theoreticaI and computational problems arising in this type of cdculation, such .ts the applicability of the Helimsnn-Feynman theorem for various correhttion w,tvcfunctions. the difference between the energy and the induced dipole moment expressions for the polariz.rbility, the choice of the b&s set, the numerical v&e of the electric field strength and the required xcuracy of the computed energies_ In this \vork we have applied the finite-field method to study the dependence of the hydrogen molecule polarizabifity on the internuclear distance. Three wavefunctions, each corresponding to a simple modeI. were chosen: the single determinant SCF function and two pair-excitation muiticonfiguration SCF (MC SCF) functions [4] *I-\‘Ct =coIlo and
5 10 z I-kc,llo
u lo,1
The orbital symmetry is shown for the unperturbed molecule lying along the z asis, it is lower for nonzero Geld strength. We shall use the superscripts SCF. MC 1 and MC2 for molecular properties computed using the corresponding ~avefunctions. As well known_ the first of the MC SCF functions can be used to describe properly the dissociation of the molecule, uhereas the second function, *Irucz_ xcounts at least partly for all the main correlation effects, i.e. the leftri$t, angular and in-out correlation_ The MC SCF approximation has additional advan@es, demonstrated using the generalized Brillouin theorem [5.6] _Thus, the MC SCF wavefunctions are appropriate for the computation of molecular properties since they satisfy the Hellmann-Feymnan theorem. In addition, for a two-electron system like Hz in contrast to the contiguration interaction apixoach there is no need to inciude other excitation schemes within the set of occupied orbit& (because these orbitak are re-optimisedj. There are numerous calculations of the hydrogen molecule polarizability going beyond the SCF level of approsimation [7--121. The exact results for a wide range of internuclear distances were determmed by KoIos and Wolniewicz [7] _ In most of the other works the polarizability was evaluated onIy for the equihbtium geometry,
R = 1.4 au. Unfortunately,
for both 189
components of the puhrir.rbility_ 0; - p-lr&cf 3nd ar_- pcrpcndianlh~r. rhe corrchttion currcctions for the tqudibrium inWrnnck3r &&~ncr‘ drr’ \er) sm.dl_ Ilk nic3ns rh3r to obt3in rcli3hlc nurncrikxii rtxifts a bctter appru\in~tion 10 the \r;l~efuncrion thtn lir’tf‘i or tk ‘*C shrruld be urcd_ 011 the other h3nd. for large intrruucle3: sep3ration the sin$r dctcrmin3nt dcscription hds coinplrtci\-_ e-g_ foraz the ti.utrce- Fock cchrmr predicts ;t bbc Itittit ior R - O” [ I-S!. Thcrefore. 711 anzdysis of the t1itfe:~ncc.s bet\\ecn cfsCr . cZ-‘K-I_Qw2 md _ riw exact pokriz~hiliry is eimphficd by inspection of their dspcndcnct on rhc inrsrrurclr.tr distLt1tce_ A bGs set of 10 s (contracted to 6 s) .mJ -I p g;lussirenorbit& r\zs considered 10 bc ~deqn;lte. This basis 1 i&is lor R = l-4 XI the iolhn~ ing tot4 energies: @-I- = _ 1_Ijjr<>g, [+<‘i = _ I_ 15 IS(j-3 dt,; I._\lc?
= - I _16SOS-?(in au). R hich compare N cl1 with ;>ther MC SCF rcsutts tar kI1 [I-%1_At the same tin:= it is x basis flexible enough fo describe re;lsontlbiy the molecuI3r polnrizabiliry_ This is confirmed b_v computetions (for each geometry considered) \\irh x corresponding electric field kkzmt basis set It5_16] \~ithin the SCF approximurion_ fiolievcr_ for the rquilibrium 3wmerry the Jifferencts bewcrn OI:C(strtndzrd b-fiis) results for QF’ = 6_4?S zu at~da”” = 4.609 zntand the more accurate coupled fiartree-Fock values of cr, = 6379 au and ar ~4-6 I7 au [ 171 are of the same order of magnitude as the correiation contributions_ The cakuI~ions were performed using the JIOLECULE system of programs [IS] _ The pokirJbikties were obtained from rhe energy difference formula, for the ciectric field strength equal to 0.005 ZIU_ In table 1 we present the fir& results for all the internuclear separations. We obseme thzlt qualitative rrgreement with the e_uct vdues can be obtained using the rwo-determinant function \kXICI I For each oeometry both components of the polarizabiIity, ap&I and &““, are too small - in contrast to the SCF re1. sulk A much better approsimxtion is provided by q3*c_ the errors do not exceed 4% for a;\xcz and 2% for aa‘Ia- This may be regarded as a satisfying result recalling that even \k-\fc7-is a rather simple model wavefunction- The remaining discrepancy is pzrtly due to the neglect of higher excitations in the multiconfig uration expansion and partly due to the defic?encies of the basis set ued_ Both these effects are more important for the perpendicuiar component of the poIarizability_ 190
1x1summtlry, ihe t\\o-determinant wwefunction .rccounrs for the gross errors of the Hxrtree-Foci, ttpproxim&m .tt large internudca diskmces. The second function. ~~1~1~1~ en3blcs 3 balttnced descriptwn of the most significant correlation effects, yields a good estimztc of the p&rktbtlity for all the gcomc:ries considcrcd_ \Ye arc aware that I& large iuternuctear separation the term ‘Lcorrcl~tion” becomes somewhat mubiguous [IS] ; wr‘ have applied it for consistency and because it is in spite of th.rt generalIy used. Last not least, as illustrated by our results for the equilibrium internuclear distance_ much higher accuracy than achieved in these computations is required fo obtain the correlation contributions in this case_ This conclusion. though not unexpected, is r.rther dissatisfying because both MC SCF functions used recover 3 vasst portion of the correI3tion energy. This work \\asbegun during 9 short stay of h1.J. in Lund. He wants to achnowledge 31~0 some helpfui discussions with Dr. A_J. S;ldlej and finrtncial support of the Polish Academy of Sciences within the proJect MR.19.
References
111 ii.J. Werner 3nd 11’.Me>er. 3101. Phys. 3 1 (1976) 855. Pi J-E_ Gread),
G-B. Banlay and NS. Hush. Chem- Ph\ S. 23 (1977) 9f31 R-D_ Amos, MoL Phyr 35 (1978) 1765_
C~IL~IICXL [a] F_ C iementi .md A. VcdLud, Thearct_ Chmt. -xc12 7 (1967) 133.
[sl
I;- Sz.de~~icz. L. _Ad;lnuxt io .md -1.J. S.tdlq_ C hem. 6 i ( 1939) 5-W. [ 9 j 1i-mVq rr, Chrm. I’ll> s I7 (I 976) 77. [ 101 U. Kaidor. J. Chcm. Ph\s. 61 (1975) 163-f. [ 1 I] T- IragAi and A. StlAn. Chem_ Ph) s_ Lcrters 52 ( 1977) 530. [ 121 R-J. B~rtlctt. J.C. Bcllum dnd L.J. ikmdas. lnrcrn. J. Ountum Chem_ 7s t 197314l9. I’h> s Letren
I’IIYSICS LCT-i-I RS
I November
1979
[ 13 j \I. Jazxuisi.i Jnd A.J. Sadiq, intern J. Quntum them. 1 i (1957) 233. f 131 G. DJS .md AC. \\.thl. J. Chem. Ph? s. 44 (1966) S7_ [ 151 A.J. S.IdleJ. Chem. Ph> s. Jxtters 47 (1977) 50. [ 161 A.J. S~dlq. unpublished rcsuirs. [ 171 X.J. Sailej .md \V_T_R+vux. UoL Dh>s_ 35 (3978) 101. [IS] J. Ahniof, Mii’ Rept. 73-19, instiiute of Ph)sics, tini\erstt> of Stockholm (I 97-l); J. Almliif, B. ROOS.md P. Stegb.dtn. Comput Chem 7 (1975) 89. [ 191 R. \tc\\an~. in- The ner\ \\orld of quantum chemistry. Proceedings of the Second interrationd Congress of Qu~nrum Chcmistn. NW Orle.ms (1976) p_ 3.
6T. number
CIfII.\IIC_AL PHYSICS
I
1 Norember
LE-ITERS
1979
FINITE-FIELD MC SCF STUDY OF THE HYDROGEN MOLECULE POLARIZABILITY
Rcceired
20 JoI:. I979
The polxiubilit~ of the II2 molecule IS cornputrd usmg the tinitc-ticld method .md rclattwl) simple multiconfiiuraticn ‘33‘ ~~.tvefttnctions_ .An.d~ sing the drpcndcncc of the polkzabdity on the internuclear dntnnce It is shoxbn that rxithin the chnxn appruGm.!tmn the correhtlon contrtbutions to the poIariz.tbiIit> can be cstimJted niten they become signifktnt.
The finite-field perturbation theory 11~sbeen widely used to study .ttomic and molecular pohrriz~bilities. In particular, It was recently applied to analyse the correlation corrections to the static dipole polarizsbilities of small molecules [ I--3] _ We refer to these papers for d thorough discussion of the theoreticaI and computational problems arising in this type of cdculation, such .ts the applicability of the Helimsnn-Feynman theorem for various correhttion w,tvcfunctions. the difference between the energy and the induced dipole moment expressions for the polariz.rbility, the choice of the b&s set, the numerical v&e of the electric field strength and the required xcuracy of the computed energies_ In this \vork we have applied the finite-field method to study the dependence of the hydrogen molecule polarizabifity on the internuclear distance. Three wavefunctions, each corresponding to a simple modeI. were chosen: the single determinant SCF function and two pair-excitation muiticonfiguration SCF (MC SCF) functions [4] *I-\‘Ct =coIlo and
5 10 z I-kc,llo
u lo,1
The orbital symmetry is shown for the unperturbed molecule lying along the z asis, it is lower for nonzero Geld strength. We shall use the superscripts SCF. MC 1 and MC2 for molecular properties computed using the corresponding ~avefunctions. As well known_ the first of the MC SCF functions can be used to describe properly the dissociation of the molecule, uhereas the second function, *Irucz_ xcounts at least partly for all the main correlation effects, i.e. the leftri$t, angular and in-out correlation_ The MC SCF approximation has additional advan@es, demonstrated using the generalized Brillouin theorem [5.6] _Thus, the MC SCF wavefunctions are appropriate for the computation of molecular properties since they satisfy the Hellmann-Feymnan theorem. In addition, for a two-electron system like Hz in contrast to the contiguration interaction apixoach there is no need to inciude other excitation schemes within the set of occupied orbit& (because these orbitak are re-optimisedj. There are numerous calculations of the hydrogen molecule polarizability going beyond the SCF level of approsimation [7--121. The exact results for a wide range of internuclear distances were determmed by KoIos and Wolniewicz [7] _ In most of the other works the polarizability was evaluated onIy for the equihbtium geometry,
R = 1.4 au. Unfortunately,
for both 189
components of the puhrir.rbility_ 0; - p-lr&cf 3nd ar_- pcrpcndianlh~r. rhe corrchttion currcctions for the tqudibrium inWrnnck3r &&~ncr‘ drr’ \er) sm.dl_ Ilk nic3ns rh3r to obt3in rcli3hlc nurncrikxii rtxifts a bctter appru\in~tion 10 the \r;l~efuncrion thtn lir’tf‘i or tk ‘*C shrruld be urcd_ 011 the other h3nd. for large intrruucle3: sep3ration the sin$r dctcrmin3nt dcscription hds coinplrtci\-_ e-g_ foraz the ti.utrce- Fock cchrmr predicts ;t bbc Itittit ior R - O” [ I-S!. Thcrefore. 711 anzdysis of the t1itfe:~ncc.s bet\\ecn cfsCr . cZ-‘K-I_Qw2 md _ riw exact pokriz~hiliry is eimphficd by inspection of their dspcndcnct on rhc inrsrrurclr.tr distLt1tce_ A bGs set of 10 s (contracted to 6 s) .mJ -I p g;lussirenorbit& r\zs considered 10 bc ~deqn;lte. This basis 1 i&is lor R = l-4 XI the iolhn~ ing tot4 energies: @-I- = _ 1_Ijjr<>g, [+<‘i = _ I_ 15 IS(j-3 dt,; I._\lc?
= - I _16SOS-?(in au). R hich compare N cl1 with ;>ther MC SCF rcsutts tar kI1 [I-%1_At the same tin:= it is x basis flexible enough fo describe re;lsontlbiy the molecuI3r polnrizabiliry_ This is confirmed b_v computetions (for each geometry considered) \\irh x corresponding electric field kkzmt basis set It5_16] \~ithin the SCF approximurion_ fiolievcr_ for the rquilibrium 3wmerry the Jifferencts bewcrn OI:C(strtndzrd b-fiis) results for QF’ = 6_4?S zu at~da”” = 4.609 zntand the more accurate coupled fiartree-Fock values of cr, = 6379 au and ar ~4-6 I7 au [ 171 are of the same order of magnitude as the correiation contributions_ The cakuI~ions were performed using the JIOLECULE system of programs [IS] _ The pokirJbikties were obtained from rhe energy difference formula, for the ciectric field strength equal to 0.005 ZIU_ In table 1 we present the fir& results for all the internuclear separations. We obseme thzlt qualitative rrgreement with the e_uct vdues can be obtained using the rwo-determinant function \kXICI I For each oeometry both components of the polarizabiIity, ap&I and &““, are too small - in contrast to the SCF re1. sulk A much better approsimxtion is provided by q3*c_ the errors do not exceed 4% for a;\xcz and 2% for aa‘Ia- This may be regarded as a satisfying result recalling that even \k-\fc7-is a rather simple model wavefunction- The remaining discrepancy is pzrtly due to the neglect of higher excitations in the multiconfig uration expansion and partly due to the defic?encies of the basis set ued_ Both these effects are more important for the perpendicuiar component of the poIarizability_ 190
1x1summtlry, ihe t\\o-determinant wwefunction .rccounrs for the gross errors of the Hxrtree-Foci, ttpproxim&m .tt large internudca diskmces. The second function. ~~1~1~1~ en3blcs 3 balttnced descriptwn of the most significant correlation effects, yields a good estimztc of the p&rktbtlity for all the gcomc:ries considcrcd_ \Ye arc aware that I& large iuternuctear separation the term ‘Lcorrcl~tion” becomes somewhat mubiguous [IS] ; wr‘ have applied it for consistency and because it is in spite of th.rt generalIy used. Last not least, as illustrated by our results for the equilibrium internuclear distance_ much higher accuracy than achieved in these computations is required fo obtain the correlation contributions in this case_ This conclusion. though not unexpected, is r.rther dissatisfying because both MC SCF functions used recover 3 vasst portion of the correI3tion energy. This work \\asbegun during 9 short stay of h1.J. in Lund. He wants to achnowledge 31~0 some helpfui discussions with Dr. A_J. S;ldlej and finrtncial support of the Polish Academy of Sciences within the proJect MR.19.
References
111 ii.J. Werner 3nd 11’.Me>er. 3101. Phys. 3 1 (1976) 855. Pi J-E_ Gread),
G-B. Banlay and NS. Hush. Chem- Ph\ S. 23 (1977) 9f31 R-D_ Amos, MoL Phyr 35 (1978) 1765_
C~IL~IICXL [a] F_ C iementi .md A. VcdLud, Thearct_ Chmt. -xc12 7 (1967) 133.
[sl
I;- Sz.de~~icz. L. _Ad;lnuxt io .md -1.J. S.tdlq_ C hem. 6 i ( 1939) 5-W. [ 9 j 1i-mVq rr, Chrm. I’ll> s I7 (I 976) 77. [ 101 U. Kaidor. J. Chcm. Ph\s. 61 (1975) 163-f. [ 1 I] T- IragAi and A. StlAn. Chem_ Ph) s_ Lcrters 52 ( 1977) 530. [ 121 R-J. B~rtlctt. J.C. Bcllum dnd L.J. ikmdas. lnrcrn. J. Ountum Chem_ 7s t 197314l9. I’h> s Letren
I’IIYSICS LCT-i-I RS
I November
1979
[ 13 j \I. Jazxuisi.i Jnd A.J. Sadiq, intern J. Quntum them. 1 i (1957) 233. f 131 G. DJS .md AC. \\.thl. J. Chem. Ph? s. 44 (1966) S7_ [ 151 A.J. S.IdleJ. Chem. Ph> s. Jxtters 47 (1977) 50. [ 161 A.J. S~dlq. unpublished rcsuirs. [ 171 X.J. Sailej .md \V_T_R+vux. UoL Dh>s_ 35 (3978) 101. [IS] J. Ahniof, Mii’ Rept. 73-19, instiiute of Ph)sics, tini\erstt> of Stockholm (I 97-l); J. Almliif, B. ROOS.md P. Stegb.dtn. Comput Chem 7 (1975) 89. [ 191 R. \tc\\an~. in- The ner\ \\orld of quantum chemistry. Proceedings of the Second interrationd Congress of Qu~nrum Chcmistn. NW Orle.ms (1976) p_ 3.