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A calculation of the well depth for a pair of helium atoms
CHEMICAL PHYSICS LCI-TERS
Volume 50, number 1
A CALCULATION
15 August 1977
OF THE WELL DEPTH FOR A PAIR OF HELlUM ATOMS
Peter D. DACRE lkpartment Rcccivcd
of Chemstry.
Unwersity
of Sheffield. Sheffield S3 7HI.t UK
16 May 1977
A new method is proposed for the elimination of tile bask supcrpo~ition error in the calculation of 51mll potentials. The mctbod which involves CI IS applied to tire calculation of the well depth of I&. It includes intcratolllic dnd intraatomic correlation cffcctc simultaneously. A value for the well dtqth of 10.54 K at I? = 3.03 au is obtained, In ewcllcnt ap,reeliicnt with current tlieorcticat estimates.
1. Introduction
The He, potential surface may be regarded as consisting of three regions. A short range strongly repuIsive potential the main features of which have been successfully described by SCF methods [ 1 J. A long range attractive van der Waals interaction, which can be successfully descrlbcd using perturbation methods [2,3], and finally an intermediate region, containing the potential well, where the repulsive and attractive interactions are of comparable size. The potential well is very shallow (=0.00003 au or x 10 K) and its accurate theoretical calculation has proved difficult. This has been largely due to the “basis set superposition” errors [4] in the correlation energy which are of the Same rnagnilude as the well depth. Thus a potential is often calculated by subtraction of a reference energy such as the energy corresponding to large or infinite separation of the two components. However if the energies are calculated
using “1ocalIy” incomplete basis sets then small errors appear because the basis set used for the interacting system will be less deficient than that used for the reference calculation. In other words each of the interacting atoms can make partial use of the basis orbitals centred on the other in order to improve its “own” wavefunction. Such effects obviously decrease with separation thereby introducing an error into the calculated potential. These basis set superposition effects are of course distinct from the changes in intra-atomic energies which occur because of the physical presence
of the second system. Previous calculations have attempted to deal with this problem either by calculating the interaction directly [5,4], or by carrying out the calculation in such a way that the superposition effect is reduced 141. In this paper we describe a method wcreby the same basis superposition error is introduced into both the pair calculation and the reference calculation, thcrcby chminating its effect by cancellation. In section 2 of the paper we describe this method, section 3 describes the results obtained, and section 4 discusses the relationship of this work to the previous calculations.
2. Method of calculation
The method is straightforward and is similar in some respects to the ghost orbital approach for SCF polarizabilities [7]. First we carry out, for Iiez at a given separation, an SCF calculation followed by a Cl calculation which includes all single and double cxcitations from the SCF ground state. Following this an cquivalcnt reference calculation, used in order to obtain the potential for the particular separation, 1s calried out on a single atom, but using exactly the same orbitals as for the pair calculation (i.e. the second atom is removed but not its orbltals). In this way an equivalent basis set superposition effect is expected for both the interacting and reference state. This term should cancel m the evaluation of the potentia1. The
CI cafcufations are carried out by the method decribed m an earlier paper (‘I]. In &riving our potential the usual practise of taking a sum of an SCF contribution, together witfl a correlation term has been followed. Prci~~x~ix~rycalculations indzcated that the correlation energy term remained insensitive to improvement of the SCF determintint, once a reasonabJy accurate SCF wavefunction had been obtained. We have therefore calculated the potential by using our calculated correlation energy term together with the accurate SCF potential given by McLaughlin and Schaefer [S]. Their SCF potential is slightly more accurate than that obtained from the CT0 basis ~ur~ctions used here. WC now discuss the evaluation of the correlation energy term for the reference state. Mad the CI calcul&ion on the interacting system been complete, then a proper choice of the noninteracting wavefunct 1011 woutd be a simple product of the ~vavefu~ct~ons of the two He atoms, so that the correlation energy term would be twice that for a sin&z atom. However with limited CI, it is ncce,uary to remove from this product terms whicfl correspond to triple and higher cxcitations. We sflafl refer to the resulting function as a double excitation rcferencc function. Let 9X, a smglet wavefunction for :Itotn X, which contains single and double excttations, be written as \z’X =$+C;$;+C$$,
0)
where I)$ is the normalized base (SCF) determinant, $k is the part (norrnalizd) of the wavcfunction containing i-fold excitations, and tfle Czx’s CorresporIdin~ coefficients_ A correct double excitation reference wavefunction for two atoms A and I3 would be *AIs =IL~~~~c~~~(~r~--~~3/~)+c~JI$J,~. Once we fllivc valrles for the Hi?
=iJl$%x$$dq
(2)
l7iF?triK elements
i=O,IJ;
X=A,B
(3)
(wfiere St is tflc flaniiItonian operator for atom X), and the Ci5 , we can determine the conelation energy term. The sunplest method of obtaining a double excitation reference state COrrCli~tiOll energy is to use the cocff
15
CHEMIC’ALPHYSICS LETTERS
Volume 50, number 1
August
1977
drupfe excitation reference state, not for the double excitation reference state.) In order to obtain a more accurate function we took advantage of the method used to solve the CI secular probIem. The CI ~igenfunct~ons are obtained using Davidson’s algorithm [IO]. At each iteration a correction function is obtained, and the solution then expressed as a linear combination of the correction functions obtained up to that Iteration. Thus after N iterations \fr, is given by Iv (4) where +x(t%) is the norInali~ed Mh correction function and oar its coefficient. In an obvious notation we write &&)
= N [G:: f d$&G
-i-d$%$@k)] tr
(S)
N being a normalization
factor. WC have used the functions y!&(k), k=O, ...N;as a basis set for the variational determination of the optimum $k and Cl’ for the double excitation reference function. (Tflc required matrix elements being easily obtained from the CI program_) The energy expression which results from such an expansion is a rational cfuartic in the various linear expansion coefficients, and its value was mininzized easily and rapidIy using Fletcher’s rninimi~~tion method [if ](we used subroutine VA 10 A).
3. AppIi~tion
of the method
All calculations were made usmg gaussian expansions of Slater orbitafs as the basis functions*. The Slater basis consisted for each atom, of the set of 5 s orbitals obtained by Cfementi [i3], together with a single p and/& d orbital. The SCF potential was investigated first. In table 1 we show the results obtained using various expansion lengths for the s orbitals. As might be expected the results tend to the accurate Slater basis resufts as the tails of the atomic wavefunctiorls are improved by using longer expansions. The basis superposition effects are small for SCF wavefunctions of this accuracy, being about I K for the 3 GTO expansion, and rapidly decreasing as the expansion * All CT0 cupansions wcrc taken from ref. [ 121.
Volume 50. number 1
IS
CHEMICAL PIIYSICS LBJXCRS
Auguct
1977
Table 1 SW potential at R = 5.6 bobr for He2 using various eupansian lengths for the s orbitals”). Hartrcc-rock energy for He is -2.8616799. Corresponding SW potentials are ret. [7] (9.24 k), ref. [4 1 (9.21 K)
Slatcr exponent of the correlation contribution to the potential at 5.6 au. Not surprisingly, the p exponent leading to the lowest potential (1.22) is somewhat re-
He SCF energy in same basis El (au) __ _--_ _ __ - ---__-_ - 2.8591631 b) -5.7 182995 --2.859 1999 -5.7 183729 -2.8602507 -5.7204734 -2.86 10545 -5.7220801 __ -_-_- _-----_----
our optimiycd exponents with those of ref. [5] is quite interesting. They Included both 2p and 3p functions, each with the same exponent. At relatively short separations (3.0 au) we obtained essentially the same 21, exponent as McLaughlin and Schaefer, whilst at separations greater than 5.0 au WC obtain a smal!cr exponent. Howcvcr closer examination indicates that our 2p function now has a very strong overlap ot’0.77 with their 3p function. Thus, quite reasonably, the functions most important for the description of interatomic interactions when coupling with the mtraatomic correlations is neglected, continue to play the same role when this is included. Finally calculations were made at a number of scparations, using a basis which included both p and d orbit:&. (d orbitals alone do not contribute to a particularly substantial well, but a large coupling effect is observed when both p and d orbitals arc included.) The results quoted in table 3 Indicate a value of 10.54 K for the well depth at R = 5.73 au (3.03 A). A number of observations arc rclcvant at this point. Firstly, unless the pair basis is used to calculate the correlation energy of the double cxcltation refcrencc state unacceptable basis set superposition errors will occur. (The error assumes the following values for a calculation of the He atom correlation energies usmg one of our basis sets: R = 3.0 au error = 459 K; R = 5.6~ error = 47 K;fi = 10.0 au error = 0.0 K.) These errors are much larger than the corresponding SCF errors. Secondly the “relaxation” of the noninteracting wavefunction in obtaining the optimum double excita-
--------
____-
--
Length of gaussian expansion ___-3 4 S 6 ___--
-_ _- - ---
He2 SW energy & (au)
-
A-2 - 26, (K) -_ - -_ 8.44 7.86 8.84 9.13
a) Basis set, 5 1s orbitals per atom, exponents from ref. 1131, plus STO-3CTO to 2p exponent 1.6 and 3d exponent 1.3. b, Energy using atom ccntrc orbit& only 1s - -2.859 1374 which yields a potential 1 K smaller than obtamed with the two ccntre basis set.
length is increased. At the same time WC found tllat the correlation energy contribution to the well was insensitive to these smiill clranl~es in the atonlic wavefunction. In our study of the correlation effects we used 4 GTO cxpanslons of p and d orbit&. The p and d exponents were optimized, thus the p orbital exponent was optimized (in the absence of the d orbital) and table 2 indicates the dependence on the Table 2 Correlation contribution to He2 well depths dt R = 5.6 bohr for various values of the p orbital exponent - _ -_-_ _ __A. ___---_-___--__ Potentidl CE(tle2) Cl’(I le) CE(lieX2) p exx 10 c) x 10”) x lob) ponent (K) -_ --__ --__ -_ - - -._ -0.3 0.6 1.0 1.15 1.22 1.3 1.6 2.0 3.0 4.0
-0.333496 -0.337607 -0.387267 -0.425921 -0.447307 -0.473634 -0.577016 -0.680394 -0.685799 -0.566429
s osbitats -0.333208 only _-___-__.-
-0.167478 -0.169511 -0.194457 -0.2140t3 --0.224725 -0.238203 -0.290665 -0.34297 1 -0.345025 -0.284574
-0.333548 -0.337573 -0.386881 -0.425482 --0.446863 --0.473 198 -0.576672 - 0.680184 -0.685762 -0.566443
al.64 -- 1.07 -12.19 -13.86 -14.00 - 13.76 -10.86 -- 6.63 - 1.16 +0.44
moved from the value of 2.6 most important m the description of atomic correlation. The comparison of
Table 3 Potential near the well for II22 _- _ -----_-R (hohr)
___-_ -0.167332 - .-
-0.333269 -
__ __
+ 1.69
__ ____
a) CE(Hq) correlation energy obtained for He2 on inclusion of single and double c’tcitations. b, CE(He) correlation energy for full CI on He using 11~ b&is. c, CE(HcX2) correlation energy obtained from the double cxcitation reference wavefunction.
i’lcctron
repulsion (K)“)
correlation (K)
- ___ __._. -_-_-----
5.4 5.6 5.73 5.8 10.0 _-.--_--
--__a
--.____
SCF
J 5.03 9.24 6.70 bj
5.66 0.00 ._-
a) Taken from ref. [ 5 I.
_ .--
-23.76 -19.55 -17.24 -16.12 -0.64 _______ _-_-_ b,
Total (K)
- -_
- --
-8.72 -10 31 -10.54 - 10.46 -0.64 ---
InterpolaWL 149
Volun~e 50, number 1
tion reference function is necessary ii accurate results are to be obtained_ At the potential minimum its neglect would increase the well depth by about 0.7 K wfulst its neglect at R = 10 au where it has almost the same value would destroy the general agrcemcnt WC obtain with the accepted value of about 0.5 K for the long-range potential calculated from perturbation thcory [2,31. A comparison of this preliminary result with if?cent cxperimcntal results is very encouraging. Burgmans et al. [ 141 on the basix of differential elastic scattering measurements estimate a well depth in the range 10.3- 10.7 K at a separation between 2.96 and 2.98 A, whilst Chapman [15] from an investigation of the tcmpcrature dependence of the relaxation time in dilute helium g;~s, obtained the best fit to the expcrimental results by using a Bruch--McGee [16] type potential with a well depth of 11.5 K and a separation of 3.0238 A.
4. Relation to previous work As indicated earlier previous calculations have attempted to deal with the basis extension effect in two ways. McLaughlin and Schaefer [S] and Bertoncini and Wahl [6] approached the problem by using localized orbilals for a Cl in which only those configurations introducing interatomic correlation were included. Unfortunately this approach neglects the change in intra-atomic correlation energy which occurs with changing separation, so that the well depth is exaggerated by about 7%. Bcrtoncini and Wahl [8] attempted to calculate the variation of the intra-atomic correla&ion directly, but although they obtained an cstimatc which yields a very satisfactory well depth, the method is unsatisfactory in that previous good agreement with the long range part of the potential is destroyed. Perhaps a particularly significant difference between the results obtained by our method and those obtained by the above methods is indicated by the results given in table 4 which arc for calculations employing s orbltals only. We obtain a completely repulsive potentlal in contrast to the attractive potential which rcsults if only interatomic terms are included_ Indeed it may be that lhe terms which arise from the s orbital only basis calculation contain the bulk of the intraatomic variation with separation. Certainly the differI50
15 Au_glst 1977
CHEMICAL PHYSICS LE’I-I-ERS T&k 4 Electron correlation s orbital basis ------R (au)
Correlation ---__-
___. 3.0 A.6 5.6 10.0 ______
to the potentiiil
contributions
_ --_-_--
this
___-__--_ energy term (K) ----
calculation
182.4 13.6 1.6 0.0 -----_-_---------------
-
from 211
-------
ref. [Sj -31.2 -1.7 -0.2 0.0
ence between the two results in table4 for R = 5.6 is close to difference between the final well depths obtained from the two methods. A second approach by Liu and McLean 141 include both mteratomic and intra-atomic effects simultaneously working with two sets of natural orbitals, one of which was chosen to represent the dominant intraatomic correlation effects (and hence not containing the minor contnbutions which lead to the superposition error) and the other to represent the interatomic correlation terms. They obtained a well depth of 9.23 K, rather smaller than the expcrimcntal estimates.
5. Conclusions The method outlined in this paper is a new approach to the technically diflicult problem of calculating the helium well depth. These preliminary results show cxceilent agreement with the most recent scattering experiment results. In order to be confident of this agreement It will bc necessary to extend the calculations in two ways. Firstly the importance of basis truncation must be invcstigatcd: initial results indicate that our simple basis is in fact quite adequate. Secondly the influcnce of triple and quadruple excitation terms must be assessed: work on this is currently in progress. Notwithstanding thcsc two factors it would appear from the agreement with the experimental results that the method outlined here is an effective method for the accurate calculation of the van dcr Waals interaction between two He atoms, and therefore since that system has cne of the smallest of wells by implication it will be effective for the calculation of van der Waals interactions between closed shell systems in general.
Volume
50, number
1
CHEMICAL
PHYSICS
Finally these results lend support LOa value of the well depth of about 10.6 K, but at a slightly larger separation ttlan has been usually accepted, viz. 3.02 au.
Acknowledgement The author wishes to thank the S.R.C. for the grant of computer time on the Rutherford 360/195. The author also wishes to acknowledge the USCof the ATMOL programs provided for use on the 360/195.
References ! 11 T.L. Gilbert and A.C. Wahl, 3. Chem. Phys. 47 (1967) 3425. [2] P.W. Langhoff, R.C. Gordon and M. Karpius, J. Chcm. Phys. 55 (1971) 2126. [3 1 G. Starkschall and R.G. Gordon, J. Chem. Phys. 54 (1971) 663.
LETTERS
15 August
1977
141 l3. Liu and A.D. McLean, J. Chcm. Phys. 59 (1973) 4557. r51 D.R. McLaughlin and III. Schaefer III, Chcm. Phys. Letters 12 (1971) 244. 161 P.J. Uertoncini and A.C. Wi~hl, Phys. Rev. I.cttcr\ 25 (1970) 991. :71 N.S. Ostlund and D.L. Mcrrifield, Chem. Phys Letters 39 (1976) 612. PI P.J. Bertoncini and A.C. Wahl, J. Chcm. Phys. 58 (1972) 1259. P.D. D,icre, Theorct. Chim. Acta 43 (1976) 197. E.R. Davidson, J. Camp. Phys. 17 (1975) 87. K. Fletcher, Fortran Subroutines for Minimitition by Quasi-Newton Methods, Publ. AERE-R-7 125, Harwell, UK (1972). 1121 R.l-. Siewart, J. Chem. Phys. 52 (1970) 431. 1131 E. Clementi, IBM J. Rcs. Develop. 9 (1965) 2, suppl. I141 A.L.J. Rurgmans. J.M. Ihrrar and Y.T. Lee, 1. Chem. Phys. 64 (1976) 1345. 1151R. Chapn?an, Phys. Rev. /L. 12 (1975) 2333. 11’51L.W. 13ruch and I.J. McGee, J. Chum. Phys. 52 (1970) 5884.
151
Volume 50, number 1
A CALCULATION
15 August 1977
OF THE WELL DEPTH FOR A PAIR OF HELlUM ATOMS
Peter D. DACRE lkpartment Rcccivcd
of Chemstry.
Unwersity
of Sheffield. Sheffield S3 7HI.t UK
16 May 1977
A new method is proposed for the elimination of tile bask supcrpo~ition error in the calculation of 51mll potentials. The mctbod which involves CI IS applied to tire calculation of the well depth of I&. It includes intcratolllic dnd intraatomic correlation cffcctc simultaneously. A value for the well dtqth of 10.54 K at I? = 3.03 au is obtained, In ewcllcnt ap,reeliicnt with current tlieorcticat estimates.
1. Introduction
The He, potential surface may be regarded as consisting of three regions. A short range strongly repuIsive potential the main features of which have been successfully described by SCF methods [ 1 J. A long range attractive van der Waals interaction, which can be successfully descrlbcd using perturbation methods [2,3], and finally an intermediate region, containing the potential well, where the repulsive and attractive interactions are of comparable size. The potential well is very shallow (=0.00003 au or x 10 K) and its accurate theoretical calculation has proved difficult. This has been largely due to the “basis set superposition” errors [4] in the correlation energy which are of the Same rnagnilude as the well depth. Thus a potential is often calculated by subtraction of a reference energy such as the energy corresponding to large or infinite separation of the two components. However if the energies are calculated
using “1ocalIy” incomplete basis sets then small errors appear because the basis set used for the interacting system will be less deficient than that used for the reference calculation. In other words each of the interacting atoms can make partial use of the basis orbitals centred on the other in order to improve its “own” wavefunction. Such effects obviously decrease with separation thereby introducing an error into the calculated potential. These basis set superposition effects are of course distinct from the changes in intra-atomic energies which occur because of the physical presence
of the second system. Previous calculations have attempted to deal with this problem either by calculating the interaction directly [5,4], or by carrying out the calculation in such a way that the superposition effect is reduced 141. In this paper we describe a method wcreby the same basis superposition error is introduced into both the pair calculation and the reference calculation, thcrcby chminating its effect by cancellation. In section 2 of the paper we describe this method, section 3 describes the results obtained, and section 4 discusses the relationship of this work to the previous calculations.
2. Method of calculation
The method is straightforward and is similar in some respects to the ghost orbital approach for SCF polarizabilities [7]. First we carry out, for Iiez at a given separation, an SCF calculation followed by a Cl calculation which includes all single and double cxcitations from the SCF ground state. Following this an cquivalcnt reference calculation, used in order to obtain the potential for the particular separation, 1s calried out on a single atom, but using exactly the same orbitals as for the pair calculation (i.e. the second atom is removed but not its orbltals). In this way an equivalent basis set superposition effect is expected for both the interacting and reference state. This term should cancel m the evaluation of the potentia1. The
CI cafcufations are carried out by the method decribed m an earlier paper (‘I]. In &riving our potential the usual practise of taking a sum of an SCF contribution, together witfl a correlation term has been followed. Prci~~x~ix~rycalculations indzcated that the correlation energy term remained insensitive to improvement of the SCF determintint, once a reasonabJy accurate SCF wavefunction had been obtained. We have therefore calculated the potential by using our calculated correlation energy term together with the accurate SCF potential given by McLaughlin and Schaefer [S]. Their SCF potential is slightly more accurate than that obtained from the CT0 basis ~ur~ctions used here. WC now discuss the evaluation of the correlation energy term for the reference state. Mad the CI calcul&ion on the interacting system been complete, then a proper choice of the noninteracting wavefunct 1011 woutd be a simple product of the ~vavefu~ct~ons of the two He atoms, so that the correlation energy term would be twice that for a sin&z atom. However with limited CI, it is ncce,uary to remove from this product terms whicfl correspond to triple and higher cxcitations. We sflafl refer to the resulting function as a double excitation rcferencc function. Let 9X, a smglet wavefunction for :Itotn X, which contains single and double excttations, be written as \z’X =$+C;$;+C$$,
0)
where I)$ is the normalized base (SCF) determinant, $k is the part (norrnalizd) of the wavcfunction containing i-fold excitations, and tfle Czx’s CorresporIdin~ coefficients_ A correct double excitation reference wavefunction for two atoms A and I3 would be *AIs =IL~~~~c~~~(~r~--~~3/~)+c~JI$J,~. Once we fllivc valrles for the Hi?
=iJl$%x$$dq
(2)
l7iF?triK elements
i=O,IJ;
X=A,B
(3)
(wfiere St is tflc flaniiItonian operator for atom X), and the Ci5 , we can determine the conelation energy term. The sunplest method of obtaining a double excitation reference state COrrCli~tiOll energy is to use the cocff
15
CHEMIC’ALPHYSICS LETTERS
Volume 50, number 1
August
1977
drupfe excitation reference state, not for the double excitation reference state.) In order to obtain a more accurate function we took advantage of the method used to solve the CI secular probIem. The CI ~igenfunct~ons are obtained using Davidson’s algorithm [IO]. At each iteration a correction function is obtained, and the solution then expressed as a linear combination of the correction functions obtained up to that Iteration. Thus after N iterations \fr, is given by Iv (4) where +x(t%) is the norInali~ed Mh correction function and oar its coefficient. In an obvious notation we write &&)
= N [G:: f d$&G
-i-d$%$@k)] tr
(S)
N being a normalization
factor. WC have used the functions y!&(k), k=O, ...N;as a basis set for the variational determination of the optimum $k and Cl’ for the double excitation reference function. (Tflc required matrix elements being easily obtained from the CI program_) The energy expression which results from such an expansion is a rational cfuartic in the various linear expansion coefficients, and its value was mininzized easily and rapidIy using Fletcher’s rninimi~~tion method [if ](we used subroutine VA 10 A).
3. AppIi~tion
of the method
All calculations were made usmg gaussian expansions of Slater orbitafs as the basis functions*. The Slater basis consisted for each atom, of the set of 5 s orbitals obtained by Cfementi [i3], together with a single p and/& d orbital. The SCF potential was investigated first. In table 1 we show the results obtained using various expansion lengths for the s orbitals. As might be expected the results tend to the accurate Slater basis resufts as the tails of the atomic wavefunctiorls are improved by using longer expansions. The basis superposition effects are small for SCF wavefunctions of this accuracy, being about I K for the 3 GTO expansion, and rapidly decreasing as the expansion * All CT0 cupansions wcrc taken from ref. [ 121.
Volume 50. number 1
IS
CHEMICAL PIIYSICS LBJXCRS
Auguct
1977
Table 1 SW potential at R = 5.6 bobr for He2 using various eupansian lengths for the s orbitals”). Hartrcc-rock energy for He is -2.8616799. Corresponding SW potentials are ret. [7] (9.24 k), ref. [4 1 (9.21 K)
Slatcr exponent of the correlation contribution to the potential at 5.6 au. Not surprisingly, the p exponent leading to the lowest potential (1.22) is somewhat re-
He SCF energy in same basis El (au) __ _--_ _ __ - ---__-_ - 2.8591631 b) -5.7 182995 --2.859 1999 -5.7 183729 -2.8602507 -5.7204734 -2.86 10545 -5.7220801 __ -_-_- _-----_----
our optimiycd exponents with those of ref. [5] is quite interesting. They Included both 2p and 3p functions, each with the same exponent. At relatively short separations (3.0 au) we obtained essentially the same 21, exponent as McLaughlin and Schaefer, whilst at separations greater than 5.0 au WC obtain a smal!cr exponent. Howcvcr closer examination indicates that our 2p function now has a very strong overlap ot’0.77 with their 3p function. Thus, quite reasonably, the functions most important for the description of interatomic interactions when coupling with the mtraatomic correlations is neglected, continue to play the same role when this is included. Finally calculations were made at a number of scparations, using a basis which included both p and d orbit:&. (d orbitals alone do not contribute to a particularly substantial well, but a large coupling effect is observed when both p and d orbitals arc included.) The results quoted in table 3 Indicate a value of 10.54 K for the well depth at R = 5.73 au (3.03 A). A number of observations arc rclcvant at this point. Firstly, unless the pair basis is used to calculate the correlation energy of the double cxcltation refcrencc state unacceptable basis set superposition errors will occur. (The error assumes the following values for a calculation of the He atom correlation energies usmg one of our basis sets: R = 3.0 au error = 459 K; R = 5.6~ error = 47 K;fi = 10.0 au error = 0.0 K.) These errors are much larger than the corresponding SCF errors. Secondly the “relaxation” of the noninteracting wavefunction in obtaining the optimum double excita-
--------
____-
--
Length of gaussian expansion ___-3 4 S 6 ___--
-_ _- - ---
He2 SW energy & (au)
-
A-2 - 26, (K) -_ - -_ 8.44 7.86 8.84 9.13
a) Basis set, 5 1s orbitals per atom, exponents from ref. 1131, plus STO-3CTO to 2p exponent 1.6 and 3d exponent 1.3. b, Energy using atom ccntrc orbit& only 1s - -2.859 1374 which yields a potential 1 K smaller than obtamed with the two ccntre basis set.
length is increased. At the same time WC found tllat the correlation energy contribution to the well was insensitive to these smiill clranl~es in the atonlic wavefunction. In our study of the correlation effects we used 4 GTO cxpanslons of p and d orbit&. The p and d exponents were optimized, thus the p orbital exponent was optimized (in the absence of the d orbital) and table 2 indicates the dependence on the Table 2 Correlation contribution to He2 well depths dt R = 5.6 bohr for various values of the p orbital exponent - _ -_-_ _ __A. ___---_-___--__ Potentidl CE(tle2) Cl’(I le) CE(lieX2) p exx 10 c) x 10”) x lob) ponent (K) -_ --__ --__ -_ - - -._ -0.3 0.6 1.0 1.15 1.22 1.3 1.6 2.0 3.0 4.0
-0.333496 -0.337607 -0.387267 -0.425921 -0.447307 -0.473634 -0.577016 -0.680394 -0.685799 -0.566429
s osbitats -0.333208 only _-___-__.-
-0.167478 -0.169511 -0.194457 -0.2140t3 --0.224725 -0.238203 -0.290665 -0.34297 1 -0.345025 -0.284574
-0.333548 -0.337573 -0.386881 -0.425482 --0.446863 --0.473 198 -0.576672 - 0.680184 -0.685762 -0.566443
al.64 -- 1.07 -12.19 -13.86 -14.00 - 13.76 -10.86 -- 6.63 - 1.16 +0.44
moved from the value of 2.6 most important m the description of atomic correlation. The comparison of
Table 3 Potential near the well for II22 _- _ -----_-R (hohr)
___-_ -0.167332 - .-
-0.333269 -
__ __
+ 1.69
__ ____
a) CE(Hq) correlation energy obtained for He2 on inclusion of single and double c’tcitations. b, CE(He) correlation energy for full CI on He using 11~ b&is. c, CE(HcX2) correlation energy obtained from the double cxcitation reference wavefunction.
i’lcctron
repulsion (K)“)
correlation (K)
- ___ __._. -_-_-----
5.4 5.6 5.73 5.8 10.0 _-.--_--
--__a
--.____
SCF
J 5.03 9.24 6.70 bj
5.66 0.00 ._-
a) Taken from ref. [ 5 I.
_ .--
-23.76 -19.55 -17.24 -16.12 -0.64 _______ _-_-_ b,
Total (K)
- -_
- --
-8.72 -10 31 -10.54 - 10.46 -0.64 ---
InterpolaWL 149
Volun~e 50, number 1
tion reference function is necessary ii accurate results are to be obtained_ At the potential minimum its neglect would increase the well depth by about 0.7 K wfulst its neglect at R = 10 au where it has almost the same value would destroy the general agrcemcnt WC obtain with the accepted value of about 0.5 K for the long-range potential calculated from perturbation thcory [2,31. A comparison of this preliminary result with if?cent cxperimcntal results is very encouraging. Burgmans et al. [ 141 on the basix of differential elastic scattering measurements estimate a well depth in the range 10.3- 10.7 K at a separation between 2.96 and 2.98 A, whilst Chapman [15] from an investigation of the tcmpcrature dependence of the relaxation time in dilute helium g;~s, obtained the best fit to the expcrimental results by using a Bruch--McGee [16] type potential with a well depth of 11.5 K and a separation of 3.0238 A.
4. Relation to previous work As indicated earlier previous calculations have attempted to deal with the basis extension effect in two ways. McLaughlin and Schaefer [S] and Bertoncini and Wahl [6] approached the problem by using localized orbilals for a Cl in which only those configurations introducing interatomic correlation were included. Unfortunately this approach neglects the change in intra-atomic correlation energy which occurs with changing separation, so that the well depth is exaggerated by about 7%. Bcrtoncini and Wahl [8] attempted to calculate the variation of the intra-atomic correla&ion directly, but although they obtained an cstimatc which yields a very satisfactory well depth, the method is unsatisfactory in that previous good agreement with the long range part of the potential is destroyed. Perhaps a particularly significant difference between the results obtained by our method and those obtained by the above methods is indicated by the results given in table 4 which arc for calculations employing s orbltals only. We obtain a completely repulsive potentlal in contrast to the attractive potential which rcsults if only interatomic terms are included_ Indeed it may be that lhe terms which arise from the s orbital only basis calculation contain the bulk of the intraatomic variation with separation. Certainly the differI50
15 Au_glst 1977
CHEMICAL PHYSICS LE’I-I-ERS T&k 4 Electron correlation s orbital basis ------R (au)
Correlation ---__-
___. 3.0 A.6 5.6 10.0 ______
to the potentiiil
contributions
_ --_-_--
this
___-__--_ energy term (K) ----
calculation
182.4 13.6 1.6 0.0 -----_-_---------------
-
from 211
-------
ref. [Sj -31.2 -1.7 -0.2 0.0
ence between the two results in table4 for R = 5.6 is close to difference between the final well depths obtained from the two methods. A second approach by Liu and McLean 141 include both mteratomic and intra-atomic effects simultaneously working with two sets of natural orbitals, one of which was chosen to represent the dominant intraatomic correlation effects (and hence not containing the minor contnbutions which lead to the superposition error) and the other to represent the interatomic correlation terms. They obtained a well depth of 9.23 K, rather smaller than the expcrimcntal estimates.
5. Conclusions The method outlined in this paper is a new approach to the technically diflicult problem of calculating the helium well depth. These preliminary results show cxceilent agreement with the most recent scattering experiment results. In order to be confident of this agreement It will bc necessary to extend the calculations in two ways. Firstly the importance of basis truncation must be invcstigatcd: initial results indicate that our simple basis is in fact quite adequate. Secondly the influcnce of triple and quadruple excitation terms must be assessed: work on this is currently in progress. Notwithstanding thcsc two factors it would appear from the agreement with the experimental results that the method outlined here is an effective method for the accurate calculation of the van dcr Waals interaction between two He atoms, and therefore since that system has cne of the smallest of wells by implication it will be effective for the calculation of van der Waals interactions between closed shell systems in general.
Volume
50, number
1
CHEMICAL
PHYSICS
Finally these results lend support LOa value of the well depth of about 10.6 K, but at a slightly larger separation ttlan has been usually accepted, viz. 3.02 au.
Acknowledgement The author wishes to thank the S.R.C. for the grant of computer time on the Rutherford 360/195. The author also wishes to acknowledge the USCof the ATMOL programs provided for use on the 360/195.
References ! 11 T.L. Gilbert and A.C. Wahl, 3. Chem. Phys. 47 (1967) 3425. [2] P.W. Langhoff, R.C. Gordon and M. Karpius, J. Chcm. Phys. 55 (1971) 2126. [3 1 G. Starkschall and R.G. Gordon, J. Chem. Phys. 54 (1971) 663.
LETTERS
15 August
1977
141 l3. Liu and A.D. McLean, J. Chcm. Phys. 59 (1973) 4557. r51 D.R. McLaughlin and III. Schaefer III, Chcm. Phys. Letters 12 (1971) 244. 161 P.J. Uertoncini and A.C. Wi~hl, Phys. Rev. I.cttcr\ 25 (1970) 991. :71 N.S. Ostlund and D.L. Mcrrifield, Chem. Phys Letters 39 (1976) 612. PI P.J. Bertoncini and A.C. Wahl, J. Chcm. Phys. 58 (1972) 1259. P.D. D,icre, Theorct. Chim. Acta 43 (1976) 197. E.R. Davidson, J. Camp. Phys. 17 (1975) 87. K. Fletcher, Fortran Subroutines for Minimitition by Quasi-Newton Methods, Publ. AERE-R-7 125, Harwell, UK (1972). 1121 R.l-. Siewart, J. Chem. Phys. 52 (1970) 431. 1131 E. Clementi, IBM J. Rcs. Develop. 9 (1965) 2, suppl. I141 A.L.J. Rurgmans. J.M. Ihrrar and Y.T. Lee, 1. Chem. Phys. 64 (1976) 1345. 1151R. Chapn?an, Phys. Rev. /L. 12 (1975) 2333. 11’51L.W. 13ruch and I.J. McGee, J. Chum. Phys. 52 (1970) 5884.
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