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Pseudopotential studies of the water and hydrogen fluoride molecules
Volume 18, number
3
CHEMfCAL PHYSICS’LETTERS ._
f Februsry
.’
1573
:
I, :
FS~U~OPOTE~T~AL
A model pseudopotential
force
canstanfs,
and ionization
STUI?IES OF THE &AT;EiR ~~.~Y~ROGE~
is used to calculate valence electronproperties for H2Q and WF; Tkti calculated k-omcrries. potentials are in excellent agreement with the resuirs of corresponding &electron c&
CUl~tiOfl?..
1, Introduction Recently there has been ~or~s~derableinterest shown in the method of pseudopotentials, in which the core system of an atom or mofeculc is replaced by an effcctive or modified potential, leaving only the valence electrons to be explicitly treated i I, 2] ?. For a one-valence-electron system the modified potential may be expressed as 131
The model potential approach would obvious19 be the more practical for studying most molecules. SevertheIess, to date most.madel potential studies have been on atomic or diatonic one- or t~v~v~~n~e-electron systems. In the following we report the tesuits of mdc del potential calculations on the water and ths hydrogen fluoride molecules.
2. The model potential
in which
the sum runs over all core orbit& &. In eq. and is not arthogonal to’the core orbitals [3]. The ~seudo~otenti~ takes account of and, in effect, removes the orthogonali& requirement. For man~~~ence-electron systeins appro. priate generaf&tions of eq. (1) can be employed 14, S]. Although the ab initio modified potential of eq, ( 1) and its generalized many.va~enceclectrort relatives have been used in certain cases [6. 71, often the formalism is used otiy as a guideline and simple ana!ytical model p~~ent~~s are used in the ~a~cula~~ons[8--l 31.
{Ii?) Qy is a valence pseudo-orbjtal
‘+ &ded by reseakh grants to The Johns Ho‘pkirxs University from the National Scienfe Found&on anb. the National ‘Wxitutesof
‘He&h.
t Present address: Dkpartment of Chemistry, Wi&ita State Wichik, Kansas 67208, USA. : “A ~&e&y, ? RFf. f I ] is a review axticie on the method of psrudopotkn‘* . tids asapplicd,to atoms and mole&es. . .. . ‘,. ‘,
.
(,
.
.
.;
There are, several properties which a satisf3etory model potential should possess, among which are tfiar it should be nondocal, &e dominant term in the potential should go as I/r for Iarge P, and there should be a $mpte method of obtaining the necessary pararnHers+. In a recent paper a WiV model potentiai which possesses these and other desirable properties :.vaspmposed [8]. The potential is of the form
where 2 is the netcharge of the cbre system, B[ is a constant, arid Pi is a projection operatbr over the subspace of spherical harmonics of a given I quantum number. The eigenvaiues of the equation
:
C%EhilCAL
.Vqiume 18, number 3
PHYSICS LETTERS
Tabtc 1
&iodelpot&al p;trsmercr#‘) _-.-__l_
_._. -_-A...--
4
BP
0.83i9
-0.0433
0.8.5X4
-0.0395 --__-_-
%
Bp
-0.0023
0.0017
-0.0032
0.0000
al The parameters arc dete&Gned fsam esperimental energies using eq. (51. The parzxetcrs arc in atomic units. b) Experimental energies used in eq. (5) SIC from ref. i 131.
where i is the appropriate angular momentum quantum 0 for g;oilnd valence states. The mo‘, ‘def potential &as suc;essfuIfy applied to one- and twov’alence-electron systems [Sj , aid Barthelat and Durand [ 12) have recently employed it to obtain IocaIized orbit& for methane. nun~berandP=
3: The calculations To Study the molecules Hz0 and HF, we first consider the one-valence-electron systems 05+ and Fsc. Ey substituting the experimental’ground state ionization energies for different symmetries in eq. (51, the corresponding 81 values can be determined; these are given in table 1. For the purpose of determining the core-core repulsions in the two molecules, the core electrons on oxygen‘and fluorine were considered to be point charges. .Tltis approximation could be eliminated, but it is con+ sistent with the p~osop~y of complete removing the core orbitals from the calculation. Thus the eight ~a. Ience electrons of water are treaied as being in the potential field of the two protons and the 05+ modei po. tentid :. The single-center SCF method was employed to cal-
$laie
the pseudowavef~nctions.
This method is parti-
’ cufariy appropriate for three reasons.. (1) The ‘I-dependent model potential acts only on
‘functions with we&defined I-qua!n’rum*numbers; as all .the b~s~s’fun~t~~nsare centered on the oxygen atom. (2) No mu@-cent& integrals involving the mod!1 . pot&iaf ha+&to be evaluatkd.
I February 1973
basis set in the ?seudopotential calculations and determining the same valence properties, a simple and direct test of the model potential approach can be made. To determine the equi~ibriun~ geometry of water, R,_, was fmed at 1.814 bohr and OH_~_H varied to mini-snize the enerw. Then the antie was fmed and R,_, varied until a minimum was found. As the final v&e of RO_ tl was VJite close to the initially assumed value, no further minimization was performed. The bending and stretching force constants were dete~ined using appropriate po~ynorni~l fits; the ionization Fotentirll was estimated using Koopmans’ theorem.
An anaIogour
procedure
was used to detet-
mine th.: relevant properties of HF. In tables 2 and 3 the calculated valence properties are compared to both Moccia’s results and to the experimental values. The excellent agreement between the &-electron and ps~~doFotentj~ caI~nIations show that for Hz0 and NF, at feast, valence properties can be su~~essftI~y predicted via the model potential approach. in both the HF and the Hz0 calculations, the orbital eigcnvalues and eigenfunctions of the three highest occupied orbit& are in excellent agreement with the corresponding eigenvalues and eigenftinctions obtained b:r itloccia. The calculated eigenvaIues of the lowest valence Ievef is, however, too positive by appro~im3~e~y 0.3 hartree in each molecule. Whiie this difference does not appear to affect the valence properties Wed in tables 2 and 3, it does affect the total valence energy, and makes it difficult to estimate the totaj moiecular energy. This discrepancy may be at least partially assigned to the method used to calculate the electronic rep& sions. In 9 rn~~v~ence-electron system the electronic repulsion terms a;e overestimated when the appropriate integrals are evakated over pse~doorbit~s instead oi orbit& [l, 4, $1. As we have previously shown [81) the energy n&y be approximately corrected if the dIectron rep&on operator is scaled down to 0.90/r,z in integrals~over pse~doorbitals. Oxygen : and fluorke on& possess s,cbre orbit@, ,so t.h? in the present case t~~~o~res~onds to multipIy~g integrals OFthe type <$,Oill/r12 I$~$$ by.(0.90)n/4, where’n is’ the number of’s o&it+ in the integral, ” When r&s corr&iori is included iri the_c&ulatio& ‘. the valcnce,qrbitai energies tid to&I valence energies of watef,rlnd bydrog& fI&$ide m’ay be calcuiateci; thti ‘.
:-
~
.,-,.,
.’ .:.,.,.
. .
“.:I-
‘-. .,
‘.
i. ‘:
._.
. .
.
..,;:,’
(.,..
r,
(..
.
,:.:
.:.,’ ..‘..
.,_
_,, ,,
,.
-...
,.
‘,,.
2.
:.:
‘,
I,
VoIume 15, number 3
CHEMICAL PHYSICS LETTERS
1 February 1973
Table 2 Valence properties of HzOa) Property
Model potential b,
ALI-electronc)
----RO-H ‘H-O-H
1.829
Table 3
Valence propcrfies of HF ‘) Esperimental d) -
1.814
105.2
106.54
1.809 104.52
IP
0.489
0.495
0.465
I(,
0.66
0.67
0.54
Kg
0.031
0.033
0.019
4. Discussion The results presented here, together with those of Barthelnt and Durand [ 121, demonstrate that model potential calculations can yield excellent predictions of the valence properties of first row molecules. If modificaiicns are made in the evaluation of the electronic-
a) b) cj d) e)
Esperimentalb)
RnT: Kr
1.716 0.72
1.728 0.74
1.733 0.62
IP
0.62
0.64
0.60
a, b, and c of table 2. values quoted in refs. [ tS. l5 ] _
Table 4 Orbital energies and torn1 energies a,b,c) -__--__ ___~__
--I___
H20
Atl-eiectron
repulsion terms, reasonably accurate orbitat and total energies may be calculated: but these alterations do not appear to be necessary for derermininn, equilibrium geometries, bending and stretching force constants, and ionization potentials. A possible source of error in the calculations lies in the method of obtaining the model potentiai parameters. From eqs. (1) and (2) we note that the modified potential is a function of the valence pseudoorbital. For OS+, for example, the valence s pseudoorbital is close to the core, but ir. H,O the valence electrons shield one another so that the s portions of the 3A, pseudoorbital are more diffuse, dws changing the modified potential. The model potential emp!oyed in the molecular calculation should re.fIect this change: in some molecules this change in the B, parameters ma> be significant.
results are given in table 4. By adding the total valence electron energies to the appropriate core energies, total molecular energies are obtained.
HF
Model potential
a) See footnotes b) Experimental
a) Au properties except angles are in atomic units; an&s are in degrees. b) See the test, section 3, for the method of determination of these properties. c) W-electron values are from refs. [ 14, 151. d) Experimental values are from ref. [ 161.
Molecule
Property
Valence orbital energies -___-_-_______--. --
Total valence energy dj ._ _... -. _-_--_. _.__... -_--
Total energy e,
-t00.001
-1.525
-0.784
(-1.533)
(-0.751)
-0.670 (-0.643)
-0.670 (-0.643)
-24.516 -
(-loo.005)
-1.301 (-1.326)
-0.721 (-0.681)
-0.600 (-0.556)
-0.532 (-0.495)
-16.851 -
-75.962 ’ (-75.922)
AU energies are in hartrees. Values in parentheses are from all-electron calculations. See refs. [ 14, 151. The electron-repulsion terms have been reduced as indicated in the text. Includes proton-core repulsions. Total energy determined by adding the total valence energy to the core.cnergy. ref. [17].
,’
Core energies of 06* and FTC obtained
from
Volume 18, number 3
CHEMICAL PHYSICS LETTERS
In the case considered here, ihe modei Fotential only removes two electrons from +he calculation. Nevertheless, HF and H,O are certainly quite different from HN and H,C, so that even in these cases a pseudopotential or a model potential is required to prevent the collapse of valence electrons into core space. For many other molecules the number of core electrons and hence the corresponding reduction in computational difficulty, would be considerably greater. Model potential approaches should be able to successfully treat such molecules.
Acknowledgiment The author would like to thank Dr. Robert G. Parr for useful discussions.
References. [ 11 J.D. Weeks. A. Hazi and %A. Rice, Advan. 16 (1969) 283
Chem.
Phys.
1
February 1973
f2] G. Simons, Chem. Phys. Letters 12 (1971) 404. [3] G. Simons and A. Matziotti, J. Chem. Phys. 52 (1970) 2449. [4] L. Szasz, J. Chem. Phys. 49 (1968) 679. [S] J.D. Weeks and S.A. Rice, J. Chem. Phys. 49 (1968) 2741.. [6] G. KcGinn, I. Chem. Phys. 50 (1969) 1404. [7] G. M&inn, J. Chcm. Phys. 51 (1969) 5090. [S] G. Simons, J. Chem. Phys. 55 (197 1) 756. [9] W.S. Struve, Chem. Phys. Letters 7 (1970) 382. Ii01W.H.E. Schwa, Chem. Phys. Letters 10 (1971)478; Theocet. Chim. Actn 15 (1969) 235, and refercnccs therein. 1111J.N. Bardslcy, Chem. Phys. Letters i (1970) 5 17. 1121J.C. Barth&t nnd Ph. Durand, Chem. Phys. Letters 16 (1973) 63. 1131C. Moore, Nat1 Bur. Std. Circular No. 467 (1949, 1952, 19.58). [141 R. Xioccia, J. Chem. Phys. 40 (1964) 2164, 2186. 1151E. hlenna, R. hloccin and L. Randaccio, Theorct. Chim. Actn J (1966) 408. 1161 R&I. Pitzcr and R.E. Merrifield, J. Chem. Phys. 52 (1970) 4782. 1171 E. Clementi, IBM J. Rss. Develop. 9 (196.5) 2, suppi.
3
CHEMfCAL PHYSICS’LETTERS ._
f Februsry
.’
1573
:
I, :
FS~U~OPOTE~T~AL
A model pseudopotential
force
canstanfs,
and ionization
STUI?IES OF THE &AT;EiR ~~.~Y~ROGE~
is used to calculate valence electronproperties for H2Q and WF; Tkti calculated k-omcrries. potentials are in excellent agreement with the resuirs of corresponding &electron c&
CUl~tiOfl?..
1, Introduction Recently there has been ~or~s~derableinterest shown in the method of pseudopotentials, in which the core system of an atom or mofeculc is replaced by an effcctive or modified potential, leaving only the valence electrons to be explicitly treated i I, 2] ?. For a one-valence-electron system the modified potential may be expressed as 131
The model potential approach would obvious19 be the more practical for studying most molecules. SevertheIess, to date most.madel potential studies have been on atomic or diatonic one- or t~v~v~~n~e-electron systems. In the following we report the tesuits of mdc del potential calculations on the water and ths hydrogen fluoride molecules.
2. The model potential
in which
the sum runs over all core orbit& &. In eq. and is not arthogonal to’the core orbitals [3]. The ~seudo~otenti~ takes account of and, in effect, removes the orthogonali& requirement. For man~~~ence-electron systeins appro. priate generaf&tions of eq. (1) can be employed 14, S]. Although the ab initio modified potential of eq, ( 1) and its generalized many.va~enceclectrort relatives have been used in certain cases [6. 71, often the formalism is used otiy as a guideline and simple ana!ytical model p~~ent~~s are used in the ~a~cula~~ons[8--l 31.
{Ii?) Qy is a valence pseudo-orbjtal
‘+ &ded by reseakh grants to The Johns Ho‘pkirxs University from the National Scienfe Found&on anb. the National ‘Wxitutesof
‘He&h.
t Present address: Dkpartment of Chemistry, Wi&ita State Wichik, Kansas 67208, USA. : “A ~&e&y, ? RFf. f I ] is a review axticie on the method of psrudopotkn‘* . tids asapplicd,to atoms and mole&es. . .. . ‘,. ‘,
.
(,
.
.
.;
There are, several properties which a satisf3etory model potential should possess, among which are tfiar it should be nondocal, &e dominant term in the potential should go as I/r for Iarge P, and there should be a $mpte method of obtaining the necessary pararnHers+. In a recent paper a WiV model potentiai which possesses these and other desirable properties :.vaspmposed [8]. The potential is of the form
where 2 is the netcharge of the cbre system, B[ is a constant, arid Pi is a projection operatbr over the subspace of spherical harmonics of a given I quantum number. The eigenvaiues of the equation
:
C%EhilCAL
.Vqiume 18, number 3
PHYSICS LETTERS
Tabtc 1
&iodelpot&al p;trsmercr#‘) _-.-__l_
_._. -_-A...--
4
BP
0.83i9
-0.0433
0.8.5X4
-0.0395 --__-_-
%
Bp
-0.0023
0.0017
-0.0032
0.0000
al The parameters arc dete&Gned fsam esperimental energies using eq. (51. The parzxetcrs arc in atomic units. b) Experimental energies used in eq. (5) SIC from ref. i 131.
where i is the appropriate angular momentum quantum 0 for g;oilnd valence states. The mo‘, ‘def potential &as suc;essfuIfy applied to one- and twov’alence-electron systems [Sj , aid Barthelat and Durand [ 12) have recently employed it to obtain IocaIized orbit& for methane. nun~berandP=
3: The calculations To Study the molecules Hz0 and HF, we first consider the one-valence-electron systems 05+ and Fsc. Ey substituting the experimental’ground state ionization energies for different symmetries in eq. (51, the corresponding 81 values can be determined; these are given in table 1. For the purpose of determining the core-core repulsions in the two molecules, the core electrons on oxygen‘and fluorine were considered to be point charges. .Tltis approximation could be eliminated, but it is con+ sistent with the p~osop~y of complete removing the core orbitals from the calculation. Thus the eight ~a. Ience electrons of water are treaied as being in the potential field of the two protons and the 05+ modei po. tentid :. The single-center SCF method was employed to cal-
$laie
the pseudowavef~nctions.
This method is parti-
’ cufariy appropriate for three reasons.. (1) The ‘I-dependent model potential acts only on
‘functions with we&defined I-qua!n’rum*numbers; as all .the b~s~s’fun~t~~nsare centered on the oxygen atom. (2) No mu@-cent& integrals involving the mod!1 . pot&iaf ha+&to be evaluatkd.
I February 1973
basis set in the ?seudopotential calculations and determining the same valence properties, a simple and direct test of the model potential approach can be made. To determine the equi~ibriun~ geometry of water, R,_, was fmed at 1.814 bohr and OH_~_H varied to mini-snize the enerw. Then the antie was fmed and R,_, varied until a minimum was found. As the final v&e of RO_ tl was VJite close to the initially assumed value, no further minimization was performed. The bending and stretching force constants were dete~ined using appropriate po~ynorni~l fits; the ionization Fotentirll was estimated using Koopmans’ theorem.
An anaIogour
procedure
was used to detet-
mine th.: relevant properties of HF. In tables 2 and 3 the calculated valence properties are compared to both Moccia’s results and to the experimental values. The excellent agreement between the &-electron and ps~~doFotentj~ caI~nIations show that for Hz0 and NF, at feast, valence properties can be su~~essftI~y predicted via the model potential approach. in both the HF and the Hz0 calculations, the orbital eigcnvalues and eigenfunctions of the three highest occupied orbit& are in excellent agreement with the corresponding eigenvalues and eigenftinctions obtained b:r itloccia. The calculated eigenvaIues of the lowest valence Ievef is, however, too positive by appro~im3~e~y 0.3 hartree in each molecule. Whiie this difference does not appear to affect the valence properties Wed in tables 2 and 3, it does affect the total valence energy, and makes it difficult to estimate the totaj moiecular energy. This discrepancy may be at least partially assigned to the method used to calculate the electronic rep& sions. In 9 rn~~v~ence-electron system the electronic repulsion terms a;e overestimated when the appropriate integrals are evakated over pse~doorbit~s instead oi orbit& [l, 4, $1. As we have previously shown [81) the energy n&y be approximately corrected if the dIectron rep&on operator is scaled down to 0.90/r,z in integrals~over pse~doorbitals. Oxygen : and fluorke on& possess s,cbre orbit@, ,so t.h? in the present case t~~~o~res~onds to multipIy~g integrals OFthe type <$,Oill/r12 I$~$$ by.(0.90)n/4, where’n is’ the number of’s o&it+ in the integral, ” When r&s corr&iori is included iri the_c&ulatio& ‘. the valcnce,qrbitai energies tid to&I valence energies of watef,rlnd bydrog& fI&$ide m’ay be calcuiateci; thti ‘.
:-
~
.,-,.,
.’ .:.,.,.
. .
“.:I-
‘-. .,
‘.
i. ‘:
._.
. .
.
..,;:,’
(.,..
r,
(..
.
,:.:
.:.,’ ..‘..
.,_
_,, ,,
,.
-...
,.
‘,,.
2.
:.:
‘,
I,
VoIume 15, number 3
CHEMICAL PHYSICS LETTERS
1 February 1973
Table 2 Valence properties of HzOa) Property
Model potential b,
ALI-electronc)
----RO-H ‘H-O-H
1.829
Table 3
Valence propcrfies of HF ‘) Esperimental d) -
1.814
105.2
106.54
1.809 104.52
IP
0.489
0.495
0.465
I(,
0.66
0.67
0.54
Kg
0.031
0.033
0.019
4. Discussion The results presented here, together with those of Barthelnt and Durand [ 121, demonstrate that model potential calculations can yield excellent predictions of the valence properties of first row molecules. If modificaiicns are made in the evaluation of the electronic-
a) b) cj d) e)
Esperimentalb)
RnT: Kr
1.716 0.72
1.728 0.74
1.733 0.62
IP
0.62
0.64
0.60
a, b, and c of table 2. values quoted in refs. [ tS. l5 ] _
Table 4 Orbital energies and torn1 energies a,b,c) -__--__ ___~__
--I___
H20
Atl-eiectron
repulsion terms, reasonably accurate orbitat and total energies may be calculated: but these alterations do not appear to be necessary for derermininn, equilibrium geometries, bending and stretching force constants, and ionization potentials. A possible source of error in the calculations lies in the method of obtaining the model potentiai parameters. From eqs. (1) and (2) we note that the modified potential is a function of the valence pseudoorbital. For OS+, for example, the valence s pseudoorbital is close to the core, but ir. H,O the valence electrons shield one another so that the s portions of the 3A, pseudoorbital are more diffuse, dws changing the modified potential. The model potential emp!oyed in the molecular calculation should re.fIect this change: in some molecules this change in the B, parameters ma> be significant.
results are given in table 4. By adding the total valence electron energies to the appropriate core energies, total molecular energies are obtained.
HF
Model potential
a) See footnotes b) Experimental
a) Au properties except angles are in atomic units; an&s are in degrees. b) See the test, section 3, for the method of determination of these properties. c) W-electron values are from refs. [ 14, 151. d) Experimental values are from ref. [ 161.
Molecule
Property
Valence orbital energies -___-_-_______--. --
Total valence energy dj ._ _... -. _-_--_. _.__... -_--
Total energy e,
-t00.001
-1.525
-0.784
(-1.533)
(-0.751)
-0.670 (-0.643)
-0.670 (-0.643)
-24.516 -
(-loo.005)
-1.301 (-1.326)
-0.721 (-0.681)
-0.600 (-0.556)
-0.532 (-0.495)
-16.851 -
-75.962 ’ (-75.922)
AU energies are in hartrees. Values in parentheses are from all-electron calculations. See refs. [ 14, 151. The electron-repulsion terms have been reduced as indicated in the text. Includes proton-core repulsions. Total energy determined by adding the total valence energy to the core.cnergy. ref. [17].
,’
Core energies of 06* and FTC obtained
from
Volume 18, number 3
CHEMICAL PHYSICS LETTERS
In the case considered here, ihe modei Fotential only removes two electrons from +he calculation. Nevertheless, HF and H,O are certainly quite different from HN and H,C, so that even in these cases a pseudopotential or a model potential is required to prevent the collapse of valence electrons into core space. For many other molecules the number of core electrons and hence the corresponding reduction in computational difficulty, would be considerably greater. Model potential approaches should be able to successfully treat such molecules.
Acknowledgiment The author would like to thank Dr. Robert G. Parr for useful discussions.
References. [ 11 J.D. Weeks. A. Hazi and %A. Rice, Advan. 16 (1969) 283
Chem.
Phys.
1
February 1973
f2] G. Simons, Chem. Phys. Letters 12 (1971) 404. [3] G. Simons and A. Matziotti, J. Chem. Phys. 52 (1970) 2449. [4] L. Szasz, J. Chem. Phys. 49 (1968) 679. [S] J.D. Weeks and S.A. Rice, J. Chem. Phys. 49 (1968) 2741.. [6] G. KcGinn, I. Chem. Phys. 50 (1969) 1404. [7] G. M&inn, J. Chcm. Phys. 51 (1969) 5090. [S] G. Simons, J. Chem. Phys. 55 (197 1) 756. [9] W.S. Struve, Chem. Phys. Letters 7 (1970) 382. Ii01W.H.E. Schwa, Chem. Phys. Letters 10 (1971)478; Theocet. Chim. Actn 15 (1969) 235, and refercnccs therein. 1111J.N. Bardslcy, Chem. Phys. Letters i (1970) 5 17. 1121J.C. Barth&t nnd Ph. Durand, Chem. Phys. Letters 16 (1973) 63. 1131C. Moore, Nat1 Bur. Std. Circular No. 467 (1949, 1952, 19.58). [141 R. Xioccia, J. Chem. Phys. 40 (1964) 2164, 2186. 1151E. hlenna, R. hloccin and L. Randaccio, Theorct. Chim. Actn J (1966) 408. 1161 R&I. Pitzcr and R.E. Merrifield, J. Chem. Phys. 52 (1970) 4782. 1171 E. Clementi, IBM J. Rss. Develop. 9 (196.5) 2, suppi.