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Variational Calculation Of The Long-Range Interaction Between Two Ground-State Hydrogen Atoms
CHEMICAL PHYSICS LElTERS
Volume 24, number 4
15 February 1974
VARIATIONAL CALCULATION OF THE LONG-RANGE INTERACTION BETWEEN TWO GROUND-STATE HYDROGEN ATOMS W. KOtOS Max Planck lmtittlte for Physics and Astroplt-vsics. and Qrcantum Ckmistry
Group,
Univcrsit~
8 Mutrick 40. Gemlany of IVarsuw. 02-093 Wurs41v, Poland *
and L. WOLNIEWICZ hmlure
of Physics, N. Copernicrrs UmTersit_v,87-100 Tonrn. Polang Received
23 November 1973
Potential energy curves far the ground state X ’Bgfand for the b “Et state of the hydrogen molecule have been cakulated for internuclear distances 6 4 R Q 12 au. The wavefunctions were represented by 60-term expansions in elliptic coordinates including the interelectronic distance. Improvement over previous results has been obtained. The computed energies are analysed from the point of view of the theory of long-range interactions.
but was identical with the expansion introduced later There has been considerable interest recently in the long-range interaction between atoms or moleto study the B lZ= state of H2 [3] _ cules, and various methods have been developed to For a given internuclear distance we expand the deal with the problem. Their applicability is usually -wavefunction in the form tested on some model systems for which accurate re\Ir = &&(1,2)+@~(2,1)] , (1) sults are available, i.e., either on the Hi ion or the H2 molecule. For Hz very accurate energies calculated by 4 = ;I /47r) exp (-&El -5&i) Peek [ 1 ] are indeed &ailable. However, in the case of H2 the accuracy of the long-range interaction energies X [exp(~~1+~~2)+(-1)s’*~~*Pexp~-~~~-~~2)J~ obtained previously by the present authors [2] seems x gi~qp&-5, (2) to be still improvable. Those computations were perwhere .$ and TJdenote the elliptic coordinates; formed in a single-precision arithrn+ic on the IBM p = r12/R; r12 and R denote the interelectronic an-d 7094 computer and due to rounding errors and to alinternuclear distance, respectively; ci, a, a, P and P most linearjy dependent basis functions for large interare variational parameters;p = 0 for g syrninitry and nuclear separations it was not possible to reach the dep = 1 for u symmetry,.and the + or_- sign in (1) refers sired accuracy. Therefore, the problem has now been to the singlet or triplet state, respectively: rev&ited and the energies have been recalculated in a ho-term wavefunctions have been constructed for d&bIe-precision arithmetic and u&g larger basis sets. ‘both the x f_zzand the b 3Gz-state by seIecting at ’ -The numerical calc@tions ha& be& carried out on R = 6 the most important terms wi& r, E & << 5. and the IBM 360/91 computer at the Max Planck Institute p < 2. In the&‘tiavefw&ionsthi? expon$ntS haYc.:‘.---.. for Physics and-Astroph$sics in Munich. 1. .beeti o&n~ed ftir R = 6; 8; 10 a$d 12 au~ass$q?’ Theexpznsion of.the wavefunctions used in the Ot= Oraqd fl F -$.-I& in$qrmgdjate _Tdues_$;f$ $e..+c: :: present work w& different from‘that employed,pre-. .. fjjal,+ms.the a-.- :>:: ~vionsly:[2] for the X.lEl and-b 3% stat& of.&; .: ’ pone&s tierti i+eq&a@:In.~e ’ &rr~e~paqsions~.tie~e suppl&ent~fl’w@ additk%l ._/: .. ..:....,... ;: __, L-_TL:._l .____:.. .:‘-y ._,._ _,..._._ ..___,;.;,i _.:, __ .- .- ‘.. ‘.. 1. ::. ,_l z1: iP~~e&dbresr-_ :,I..‘ 5:: -’ ‘ , .:._ :. .__ . .:...; .:.‘:.._ i._..‘, -::. :: ._ _.:.:4y;
Volume 24, number 4
CHEMICAL PHYSICS LETTERS
Theoretical energies for the X 1 ii
15 February 1974
Table 1 state of Hz computed in the Born-Oppenheimer
approximationa)
R
E
D
AD60-40
AD60-old
dEj:laR
6.0 6.2 6-4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 8.5 9.0 9.5 10.0 11.0 120
-i.O00834208 -1.000619694 -1.000462031 -1.000346029 -1.000260534 -1,000197372 -1.000150567 -1.000115753 -1.000089741 --1.0000702U4 -1.000055444 -1.000032076 -1.000019725 -1.000012824 -1.000008738 -1.000004502 -1.000002548
183.088 136.007 101.404 75.94s 57.181 43.318 33.046 25.405 19.696 15.408 12.168 7.040 4.329 2815 1.918 0.988 0.559
0.339 0.251 0.202 0.162 0.131 0.106 0.086 0.070 0.057 0.046 0.037 0.023 0.014 0.0 IO 0.007 0.003 0.002
4.2 3.7 3.4 2.9 2.6 1.8 1.5 1.6 0.6 0.4 0.6 0.1 0.2 0.1 -0.1
0.001245207 0.000915094 0.000672959 0.000495566 0.000365674 0.000270556 0.000200858 0.000149727 0.000112150 0.000084474 0.000064029 0.000033053 0.000017994 0.000010383 O.OOO006345 0.000002742 0.000001363
a) R, E and d E/d R in au, D and AD in cm-’ (1 au of energy = 219474.62
0.012 cm-‘).
Table 2 Theoretical energies for the b “.Zz state of Hz computed in the Born-Oppenheimer R 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 7.85 7.9 a.0 8.25 8.5 9.0 9.5
.10-o 11.0
120
’
E
D
-0.999812748 -0.999884583 -0.999934033 -0.999967485 -0.999989604 -1.000003776 -1.000012440 -1.000017347. -1.000019730 -1.000020462 -1.000020462 -1.000020405 -1.000020147 -1.000~18901 -1.000017189 -1.000013491 -1.Od0010230 -1.000007663 -1.000004320 ~~1.000002516
-41.097 -25.331 -14.478 - 7.136 - 2-282 0.829 2.730 3.807 4.330 4.491 4.491 4.478 4.422 4.148 3.773 2.961 2.245 1.682 0.948 0.552
a)R.EanddE[dRin~u,Dand~Dincm-‘.
approximationa)
dEldR 0.159 0.130 0.107 0.088 0.073 0.061 0.049 0.042 0.037 0.029 0.028 0.026 0.025 0.020 0.017
0.011 0.008 O-0060.003 0.002
0.0 0.1 0.1 0.1 0.2
0.1 0.1 0.1 0.1 0.2 0.2 OS 0.2 0.2 0.1 0.2 0.011
.-
-0.000428556 -0.oOO297125 -0.000202808 -0.000135696 -0.000088402 -0.000055433 -0.000032768 -0.000017393 -0.000007209 ~0.000000644 0.000000571 0.000001644 0.000003419 0.00000l5168 0.00000~293 O.OOtiO7132 0.00000~833 : 0.000004449~ 0.000002418 .. 0:000O01308
Volume 24, number 4
CHEMICAL PHYSICS LETTERS
D, obtained
and the im-
with the 60-term expansions,
15 February 1974
energies for the IZg and 3Zf states which yield Q(r)_. The values of Q$& have been calculated by Bowman [S] _Using these results, and assuming Qpx&, = 0 for i > 2, one gets from Q and K the values of Epol and 3E exch given in table 3. In the vicinity of the van der Waals minimum the distinction between Epol and Q is completely insignificant. In table 3 we also list the a-
provement ofD obtained by extending the expansion length from 40 to 60 terms, AD60_40, as well as the improvement over the previous results, AD6u_old. For R = 12 the improvement over Pecul’s results ]4] obtained by using the split hamiltonian matrix method is given. In the last column of tables 1 and 2 we list the derivative of the energy with respect to the inter-
symptotic exchange energies, K,,
nuclear distance calculated from the virial theorem.
formula of Herring and Flicker [6]
In table 3 we give the Coulomb defined as
KS = 0.821 X R5” exp(-2R).
and exchange energies
Q =$(3E+1E),
(3)
K=$(3E-lE),
(4)
= iE(‘) _ E(i)
where E$$ denotes the energy obtained
x +$+&+,>si+s;
in the ith or-
with respect to. permutations
= Q(i) _ E(i)
(6)
pol *
exp(--+--~2--F7~--~~2) @I
By using in (8) terms with the same powers of &, i$, ~7~and p as for the ‘Zi state, at R = 6 au the interaction energy Epolvx_ = 69.13 cm-l was obtained. However, this value could be substantially improved by repIacing some of the terms with those’that were important for the 3Zi state; The results obtained with a new 60-term expansion are listed in table3. For R = 6 au the exponents have been reoptimized. Since this resulted in a totally insignificant change of the interaction energy, for R = 8 and 10 au the same exponents as for the r Zl state were used. It is seen that the values of Epotvar_ are in a perfect agreement with those obtained from the Coulomb energies, although: q1
of the electrons bettieen the interacting systems. Note that E$‘;tiis not identical with Q(‘)which contains the exchange contribution, Q$$&,, defined as Q&
--(YE2 +&?I+ F?2)
x ~~~q~fi]p~i_
der of the perturbation theory without antisymmetrizing the wavefunction
CiIexP(-+I
* = (1/4n)f:
(5)
pot ’
(7)
Even in the vicinity of the minimum, the values of K, are seen to be fairly reliabIe. In the last column of table 3 we give the polarization energy, Epo~var., calculated directly in the present work. Similarly as for the singlet and triplet states the variational method was employed using a trial function in the form
where 3E and IE denote the energy of the triplet and singlet state, respectively. Similarly, in each order of the perturbation theory one can introduce the Coulomb energy, @), and exchange energy, I#). defined by formulas analogous to (3) and (4). . Usually it is, however, more convenient to define the exchange energy with reference to the polariza. tion energy, E$, which can be directly computed. Thus iE;i&
calculated from the
of Q$&&can readily be obtained by using the first-order polarization energy, E$,j , resulting from G$ = 1~,(1)9~(2), and the Heitler-London
The values
,
Table 3 Various components
--R.
K
_Q.
6
..
-71.00
.- 8.30
8 lo._.. 12
.-
..
1.800
- 0.556 : ..
a) R:in A, e;lergies in cALf.
.: -
_:.
of the interaction energy of two ground-state hydrogen atomsa)
:
._
3E,nclr
EPl -71.5
112.09 3.87 0.118 0.0?35
- 8.30 - 1;80 ‘-
0.556.
--.
:
112.6 3.8i 0.118 0.003j
-.;
::
:
:. -.
_. .-.
I. : ‘- . :.:
:
_ ,...__:.
-il.47
97.63 3.671 0.117. -0.do34~:
-,
_: .
~goI_“ar.
.Kas
‘.
..- ‘8.27 _’‘-_i.80.‘.-~:.. .,
.,: ...
I,
-: -->.._ _$~...
; ..-,_ ,~ L . :- .. _:’ :_: -, :, ...z. .~. .- .. :-.I__.:__ :--2.,‘45_9_ .‘,.‘-
CHEhIlCAL PHYSICS LETTERS
Volume 24, number 4
Table 4 Interaction energy of two hydrogen atoms in the b “‘;i
.R 6 8 10 --_--_--_~~
3%r
~as+~pol.“ar.
3E:i!h+Epol
41.26
26.2 -4.60 -1.68
27.6 -4.84 -1.70
-4.42
-1.68
a) R in au, energies in cm
-~--
3E var = 3Eexcll * E polvar. -
(9)
Let us focus our attention on the results for R = 8 au, i.e., on the region close to the van der Waals minimum. IfEex,,, is replaced by &, resulting from the asymptotic formula, the results are seen to be good. However, this procedure does not seem to be applicable to larger systems. If Eeschis replaced by the first-order exchange energy, 3E,$h, which in contrast to Eesch can easily be calculated. the results are still fairly good. Even a rough calculation of 3E$h would improve them significantly. Usually, however, one calculates f?(l)+ EgL where E$L is the familiar asymp totic expansion of the second-order polarization enerKY = “F3 l3,
1974
3E (IbEg]
as . _
state”) .var .
33.2
-5.6 -1.74
-1 _
the definition of EpoLv._ may certainly raise objections. In table 4 we compare the accurate interaction energies for the b 32z state, E,,..? with those which can be obtained by calculating accurately, or approximately, only some components of the total interaction energies_ Obviously, by definition we should have
E$L
15 Febtiary
R -2n _
(10)
In summing the above asymptotic expansion, the method indicated by Dalgarno and Lewis [7] may be employed. By using the Ba, values determined by Bell (81 and the Heitler-London energies, 3E(t), one gets the results listed in the last column of table 4. They are considerably worse than the previous ones, and there seems to be no way of improving them. If in a particular calculation this approach gives reasonably good intcraLrion energies. this is likely to be due to a fortuitous cancellation of errors. Still less reliable results would be obtained by taking only the first term
460
(dipole-dipole) in the expansion (IO). A comment may still be appropriate with regard to the accuracy of the present results. For large internuclear distances the accuracy seems to be very high. For R = 6 it is difficult to say anything definite. Several test runs have been made with still more extended basis sets and this either improved the energy insignificantly or made the energy unstable, i.e., addition of a new term increased the energy. In each of the final runs the 60 X 60 matrices were computed and the lowest eigenvalue was calculated for 40( 1j-60 terms. No instability of the energy has been observed for these expansions. One of the authors (W.K.) is greatly indebted to Dr. G.H.F. Diercksen for his very kind hospitality at the Max Planck Institute for Physics and Astrophysics in Munich_
References [l] J.M. Pe& J. Chem. Phys. 43 (1965) 3004. 121 W. Kolos and L. Wolniewicz, J. Chem. Phys. 43 (1965) 2429. [3] W. Kotos and L. Wolniewicz, J. Chem. Phys. 45 (1966) 509. [4] K. Pecul, Chem. Phys. Letters 9 (1971) 316. [S) J.D. Bowman, Ph. D. Thesis, University of Wisconsin, Theoretical Chemistry Institute, Report NO. WIS-TCI-
463 (1971). [6 1 C. Herring and M. Flicker, Phys. Rev- 134A (1964) 362. [7] A. Dalgamo and J-T. Lewis, Proc. Phys. Soc.(London) (1956) 57. [Sj R.J. Bell, Proc. Phys. Sot. (London) 87 (1966) 594.
69
.. ‘._
Volume 24, number 4
15 February 1974
VARIATIONAL CALCULATION OF THE LONG-RANGE INTERACTION BETWEEN TWO GROUND-STATE HYDROGEN ATOMS W. KOtOS Max Planck lmtittlte for Physics and Astroplt-vsics. and Qrcantum Ckmistry
Group,
Univcrsit~
8 Mutrick 40. Gemlany of IVarsuw. 02-093 Wurs41v, Poland *
and L. WOLNIEWICZ hmlure
of Physics, N. Copernicrrs UmTersit_v,87-100 Tonrn. Polang Received
23 November 1973
Potential energy curves far the ground state X ’Bgfand for the b “Et state of the hydrogen molecule have been cakulated for internuclear distances 6 4 R Q 12 au. The wavefunctions were represented by 60-term expansions in elliptic coordinates including the interelectronic distance. Improvement over previous results has been obtained. The computed energies are analysed from the point of view of the theory of long-range interactions.
but was identical with the expansion introduced later There has been considerable interest recently in the long-range interaction between atoms or moleto study the B lZ= state of H2 [3] _ cules, and various methods have been developed to For a given internuclear distance we expand the deal with the problem. Their applicability is usually -wavefunction in the form tested on some model systems for which accurate re\Ir = &&(1,2)+@~(2,1)] , (1) sults are available, i.e., either on the Hi ion or the H2 molecule. For Hz very accurate energies calculated by 4 = ;I /47r) exp (-&El -5&i) Peek [ 1 ] are indeed &ailable. However, in the case of H2 the accuracy of the long-range interaction energies X [exp(~~1+~~2)+(-1)s’*~~*Pexp~-~~~-~~2)J~ obtained previously by the present authors [2] seems x gi~qp&-5, (2) to be still improvable. Those computations were perwhere .$ and TJdenote the elliptic coordinates; formed in a single-precision arithrn+ic on the IBM p = r12/R; r12 and R denote the interelectronic an-d 7094 computer and due to rounding errors and to alinternuclear distance, respectively; ci, a, a, P and P most linearjy dependent basis functions for large interare variational parameters;p = 0 for g syrninitry and nuclear separations it was not possible to reach the dep = 1 for u symmetry,.and the + or_- sign in (1) refers sired accuracy. Therefore, the problem has now been to the singlet or triplet state, respectively: rev&ited and the energies have been recalculated in a ho-term wavefunctions have been constructed for d&bIe-precision arithmetic and u&g larger basis sets. ‘both the x f_zzand the b 3Gz-state by seIecting at ’ -The numerical calc@tions ha& be& carried out on R = 6 the most important terms wi& r, E & << 5. and the IBM 360/91 computer at the Max Planck Institute p < 2. In the&‘tiavefw&ionsthi? expon$ntS haYc.:‘.---.. for Physics and-Astroph$sics in Munich. 1. .beeti o&n~ed ftir R = 6; 8; 10 a$d 12 au~ass$q?’ Theexpznsion of.the wavefunctions used in the Ot= Oraqd fl F -$.-I& in$qrmgdjate _Tdues_$;f$ $e..+c: :: present work w& different from‘that employed,pre-. .. fjjal,+ms.the a-.- :>:: ~vionsly:[2] for the X.lEl and-b 3% stat& of.&; .: ’ pone&s tierti i+eq&a@:In.~e ’ &rr~e~paqsions~.tie~e suppl&ent~fl’w@ additk%l ._/: .. ..:....,... ;: __, L-_TL:._l .____:.. .:‘-y ._,._ _,..._._ ..___,;.;,i _.:, __ .- .- ‘.. ‘.. 1. ::. ,_l z1: iP~~e&dbresr-_ :,I..‘ 5:: -’ ‘ , .:._ :. .__ . .:...; .:.‘:.._ i._..‘, -::. :: ._ _.:.:4y;
Volume 24, number 4
CHEMICAL PHYSICS LETTERS
Theoretical energies for the X 1 ii
15 February 1974
Table 1 state of Hz computed in the Born-Oppenheimer
approximationa)
R
E
D
AD60-40
AD60-old
dEj:laR
6.0 6.2 6-4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 8.5 9.0 9.5 10.0 11.0 120
-i.O00834208 -1.000619694 -1.000462031 -1.000346029 -1.000260534 -1,000197372 -1.000150567 -1.000115753 -1.000089741 --1.0000702U4 -1.000055444 -1.000032076 -1.000019725 -1.000012824 -1.000008738 -1.000004502 -1.000002548
183.088 136.007 101.404 75.94s 57.181 43.318 33.046 25.405 19.696 15.408 12.168 7.040 4.329 2815 1.918 0.988 0.559
0.339 0.251 0.202 0.162 0.131 0.106 0.086 0.070 0.057 0.046 0.037 0.023 0.014 0.0 IO 0.007 0.003 0.002
4.2 3.7 3.4 2.9 2.6 1.8 1.5 1.6 0.6 0.4 0.6 0.1 0.2 0.1 -0.1
0.001245207 0.000915094 0.000672959 0.000495566 0.000365674 0.000270556 0.000200858 0.000149727 0.000112150 0.000084474 0.000064029 0.000033053 0.000017994 0.000010383 O.OOO006345 0.000002742 0.000001363
a) R, E and d E/d R in au, D and AD in cm-’ (1 au of energy = 219474.62
0.012 cm-‘).
Table 2 Theoretical energies for the b “.Zz state of Hz computed in the Born-Oppenheimer R 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 7.85 7.9 a.0 8.25 8.5 9.0 9.5
.10-o 11.0
120
’
E
D
-0.999812748 -0.999884583 -0.999934033 -0.999967485 -0.999989604 -1.000003776 -1.000012440 -1.000017347. -1.000019730 -1.000020462 -1.000020462 -1.000020405 -1.000020147 -1.000~18901 -1.000017189 -1.000013491 -1.Od0010230 -1.000007663 -1.000004320 ~~1.000002516
-41.097 -25.331 -14.478 - 7.136 - 2-282 0.829 2.730 3.807 4.330 4.491 4.491 4.478 4.422 4.148 3.773 2.961 2.245 1.682 0.948 0.552
a)R.EanddE[dRin~u,Dand~Dincm-‘.
approximationa)
dEldR 0.159 0.130 0.107 0.088 0.073 0.061 0.049 0.042 0.037 0.029 0.028 0.026 0.025 0.020 0.017
0.011 0.008 O-0060.003 0.002
0.0 0.1 0.1 0.1 0.2
0.1 0.1 0.1 0.1 0.2 0.2 OS 0.2 0.2 0.1 0.2 0.011
.-
-0.000428556 -0.oOO297125 -0.000202808 -0.000135696 -0.000088402 -0.000055433 -0.000032768 -0.000017393 -0.000007209 ~0.000000644 0.000000571 0.000001644 0.000003419 0.00000l5168 0.00000~293 O.OOtiO7132 0.00000~833 : 0.000004449~ 0.000002418 .. 0:000O01308
Volume 24, number 4
CHEMICAL PHYSICS LETTERS
D, obtained
and the im-
with the 60-term expansions,
15 February 1974
energies for the IZg and 3Zf states which yield Q(r)_. The values of Q$& have been calculated by Bowman [S] _Using these results, and assuming Qpx&, = 0 for i > 2, one gets from Q and K the values of Epol and 3E exch given in table 3. In the vicinity of the van der Waals minimum the distinction between Epol and Q is completely insignificant. In table 3 we also list the a-
provement ofD obtained by extending the expansion length from 40 to 60 terms, AD60_40, as well as the improvement over the previous results, AD6u_old. For R = 12 the improvement over Pecul’s results ]4] obtained by using the split hamiltonian matrix method is given. In the last column of tables 1 and 2 we list the derivative of the energy with respect to the inter-
symptotic exchange energies, K,,
nuclear distance calculated from the virial theorem.
formula of Herring and Flicker [6]
In table 3 we give the Coulomb defined as
KS = 0.821 X R5” exp(-2R).
and exchange energies
Q =$(3E+1E),
(3)
K=$(3E-lE),
(4)
= iE(‘) _ E(i)
where E$$ denotes the energy obtained
x +$+&+,>si+s;
in the ith or-
with respect to. permutations
= Q(i) _ E(i)
(6)
pol *
exp(--+--~2--F7~--~~2) @I
By using in (8) terms with the same powers of &, i$, ~7~and p as for the ‘Zi state, at R = 6 au the interaction energy Epolvx_ = 69.13 cm-l was obtained. However, this value could be substantially improved by repIacing some of the terms with those’that were important for the 3Zi state; The results obtained with a new 60-term expansion are listed in table3. For R = 6 au the exponents have been reoptimized. Since this resulted in a totally insignificant change of the interaction energy, for R = 8 and 10 au the same exponents as for the r Zl state were used. It is seen that the values of Epotvar_ are in a perfect agreement with those obtained from the Coulomb energies, although: q1
of the electrons bettieen the interacting systems. Note that E$‘;tiis not identical with Q(‘)which contains the exchange contribution, Q$$&,, defined as Q&
--(YE2 +&?I+ F?2)
x ~~~q~fi]p~i_
der of the perturbation theory without antisymmetrizing the wavefunction
CiIexP(-+I
* = (1/4n)f:
(5)
pot ’
(7)
Even in the vicinity of the minimum, the values of K, are seen to be fairly reliabIe. In the last column of table 3 we give the polarization energy, Epo~var., calculated directly in the present work. Similarly as for the singlet and triplet states the variational method was employed using a trial function in the form
where 3E and IE denote the energy of the triplet and singlet state, respectively. Similarly, in each order of the perturbation theory one can introduce the Coulomb energy, @), and exchange energy, I#). defined by formulas analogous to (3) and (4). . Usually it is, however, more convenient to define the exchange energy with reference to the polariza. tion energy, E$, which can be directly computed. Thus iE;i&
calculated from the
of Q$&&can readily be obtained by using the first-order polarization energy, E$,j , resulting from G$ = 1~,(1)9~(2), and the Heitler-London
The values
,
Table 3 Various components
--R.
K
_Q.
6
..
-71.00
.- 8.30
8 lo._.. 12
.-
..
1.800
- 0.556 : ..
a) R:in A, e;lergies in cALf.
.: -
_:.
of the interaction energy of two ground-state hydrogen atomsa)
:
._
3E,nclr
EPl -71.5
112.09 3.87 0.118 0.0?35
- 8.30 - 1;80 ‘-
0.556.
--.
:
112.6 3.8i 0.118 0.003j
-.;
::
:
:. -.
_. .-.
I. : ‘- . :.:
:
_ ,...__:.
-il.47
97.63 3.671 0.117. -0.do34~:
-,
_: .
~goI_“ar.
.Kas
‘.
..- ‘8.27 _’‘-_i.80.‘.-~:.. .,
.,: ...
I,
-: -->.._ _$~...
; ..-,_ ,~ L . :- .. _:’ :_: -, :, ...z. .~. .- .. :-.I__.:__ :--2.,‘45_9_ .‘,.‘-
CHEhIlCAL PHYSICS LETTERS
Volume 24, number 4
Table 4 Interaction energy of two hydrogen atoms in the b “‘;i
.R 6 8 10 --_--_--_~~
3%r
~as+~pol.“ar.
3E:i!h+Epol
41.26
26.2 -4.60 -1.68
27.6 -4.84 -1.70
-4.42
-1.68
a) R in au, energies in cm
-~--
3E var = 3Eexcll * E polvar. -
(9)
Let us focus our attention on the results for R = 8 au, i.e., on the region close to the van der Waals minimum. IfEex,,, is replaced by &, resulting from the asymptotic formula, the results are seen to be good. However, this procedure does not seem to be applicable to larger systems. If Eeschis replaced by the first-order exchange energy, 3E,$h, which in contrast to Eesch can easily be calculated. the results are still fairly good. Even a rough calculation of 3E$h would improve them significantly. Usually, however, one calculates f?(l)+ EgL where E$L is the familiar asymp totic expansion of the second-order polarization enerKY = “F3 l3,
1974
3E (IbEg]
as . _
state”) .var .
33.2
-5.6 -1.74
-1 _
the definition of EpoLv._ may certainly raise objections. In table 4 we compare the accurate interaction energies for the b 32z state, E,,..? with those which can be obtained by calculating accurately, or approximately, only some components of the total interaction energies_ Obviously, by definition we should have
E$L
15 Febtiary
R -2n _
(10)
In summing the above asymptotic expansion, the method indicated by Dalgarno and Lewis [7] may be employed. By using the Ba, values determined by Bell (81 and the Heitler-London energies, 3E(t), one gets the results listed in the last column of table 4. They are considerably worse than the previous ones, and there seems to be no way of improving them. If in a particular calculation this approach gives reasonably good intcraLrion energies. this is likely to be due to a fortuitous cancellation of errors. Still less reliable results would be obtained by taking only the first term
460
(dipole-dipole) in the expansion (IO). A comment may still be appropriate with regard to the accuracy of the present results. For large internuclear distances the accuracy seems to be very high. For R = 6 it is difficult to say anything definite. Several test runs have been made with still more extended basis sets and this either improved the energy insignificantly or made the energy unstable, i.e., addition of a new term increased the energy. In each of the final runs the 60 X 60 matrices were computed and the lowest eigenvalue was calculated for 40( 1j-60 terms. No instability of the energy has been observed for these expansions. One of the authors (W.K.) is greatly indebted to Dr. G.H.F. Diercksen for his very kind hospitality at the Max Planck Institute for Physics and Astrophysics in Munich_
References [l] J.M. Pe& J. Chem. Phys. 43 (1965) 3004. 121 W. Kolos and L. Wolniewicz, J. Chem. Phys. 43 (1965) 2429. [3] W. Kotos and L. Wolniewicz, J. Chem. Phys. 45 (1966) 509. [4] K. Pecul, Chem. Phys. Letters 9 (1971) 316. [S) J.D. Bowman, Ph. D. Thesis, University of Wisconsin, Theoretical Chemistry Institute, Report NO. WIS-TCI-
463 (1971). [6 1 C. Herring and M. Flicker, Phys. Rev- 134A (1964) 362. [7] A. Dalgamo and J-T. Lewis, Proc. Phys. Soc.(London) (1956) 57. [Sj R.J. Bell, Proc. Phys. Sot. (London) 87 (1966) 594.
69
.. ‘._