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Geometry and barriers to internal rotation in Hartree-Fock theory
Volume 2, number 4
CHEMICAL
GEOMETRY
AND IN
Department
of
PHYSICS LETTERS
BARRIERS
TO
August -1968
INTERNAL
HARTREE-FOCK
ROTATION
THEORY
K. F. FREED* ?%eOTetiCat physics,
TIte .%iversity
of i%lanchester,
ERgkZRd
Receiver3 X4 June 1968
Itis shown that in Hartree-Fock theory geometry and barriers to internal rotation can be considered in the class of one-electron operators since they are given correctly through first order in the wave fimction.
It is a well kncwn fact that one electron properties are obtained correct through first order from Hartree-Fock (HF) wave functions [l]. It is easily shown that errors in HF molecular geometry and barriers to internaf rotation are also of second order: Let the exact wave function J/ be expressed as * = (l*E2)-~(f$+EX), where r# is the HF wave function; +, I$, and x are normalized to unity; and @ and x are othrogonal. J.fA is a one electron operator and &I = CpIdlP), then because of Brillouin’s theorem @Jf Al x) - 0, and to lowest
order
A(A) = &>,
in E
- ti;jb 2 E2(&4& - (A)+).
(1)
Eq. (1) has been invoked to explain the good agreement between one electron properties which are obtained from experiment and HF calculations [l, 21. However, eq. (1) is of little practical utility since it gives no way of estimating E or (A),. Furthermore, ~2 is not necessarily small in many electron systems. However, in view of the ability to obtain good bond lengths, bond angles 133, and probably barriers to internal rotation [4] from HF theory, eq. (1) is extended to cover these properties. Because + and J/satisfy the Hellmann-Feynman theorem [5,6], &A(H) where R is a parameter
= A($ occurring
in the elec-
* NATO postdoctoral fellow. Present address: Department of Chemistry and James Franck Institute, University of Chicago, Chicago* Illinois, USA.
tronic cule, tronic brium
Hamiltonian H[?f For a diatomic moIeif R is taken to be the bond length, the elecenergy can be expanded about the equiLivalue
E=Eo+#(R-R,i2+
$ks(R-Re)3
t ..”
with a similar expansion for the HF energy. primes denote HF quantities,
, (2)
If
k
(31 (The term in k7 which $s neglected is of order g4.) The force (c?H. SRjXI Re can be estimated very
crudely as that of a “typical” HF excited s&.&e, and thus R;? - A, is correct through order E. Vnfortunately, because ((??H/%}2~ is singular, recent error bound techniques [2] carmot be used to estimate RL - Re. We can obtain a result analogous to eq. (3) for polyatomic molecules, where R represents a bond length or angle, As above we can neglect cubic and higher terms. In order to use eq. (3‘: for polyatomic molecufes, R, is the HF equilibrium value when all other coordinates are taken at their true equilibrium values. If the HF energy is expressed in terms of the HF geometry, we can either neglect the interaction force constants as being small, or interpret eq. (3) as a matrix equation which yields the same qualitative result. For an internal rotor a hose potential energy is adequately approximate’1 by the truncated potential 2 C V;2j COsR.j@, j=o
255
Volume 2, number 4
the barrier
CHEMICAL PHYSICS LETTERS
is equal to 2 Vn and [ 81
REFERENCES
and HF gives the barrier
correct through the first order. It should be noted that for any wave function (double primes) which satisfies the HellmannFeynman theorem,
(retaining only the quadratic part of the energy), and if an error bound could be calculated for a force (or torque operator), an experimental vailue of R, would give a bound for k and vice veysa. Discussions with Professor W. Klemperer R G. Parr are gratefully acknowledged.
256
August 1968
and
[l] J. Goodisman and W.Klemperer, J. Chem. Phys. 38 (1963) 721. [Z] P. Jennings and E. B. Wilson Jr., J. Chem. Phys. 47 (1967) 2130. [3] For instance: R. J. Buenker and S. D. Peyerimhoff. J. Chem. Phys. 45 (1966) 3682: S. D. Peyerimhoff, J. Chem. Phys. 47 (1967) 349. [4] For instance: W-H. Fink, D. C. Pan and L.C. Allen, J. Chem. Phys. 47 (1967) 895. [5] H. Hellmann. EinfUhrung in die Quantumchemie (F. Deuticke, Leipzig, 1937): R.P. Feynman, Phys. Rev. 56 (1939) 340. [S] R. E-Stanton, J. Chem.Phys. 36 (1962) 1298. [7J M. E.Schwartz, J. Chcm. Phys. 45 (1966) 4754. [8] J. Goodisman, J. Chem. Phys. 44 (1966) 2085: 45 (1966) 4689: 47 (1967) 334.
CHEMICAL
GEOMETRY
AND IN
Department
of
PHYSICS LETTERS
BARRIERS
TO
August -1968
INTERNAL
HARTREE-FOCK
ROTATION
THEORY
K. F. FREED* ?%eOTetiCat physics,
TIte .%iversity
of i%lanchester,
ERgkZRd
Receiver3 X4 June 1968
Itis shown that in Hartree-Fock theory geometry and barriers to internal rotation can be considered in the class of one-electron operators since they are given correctly through first order in the wave fimction.
It is a well kncwn fact that one electron properties are obtained correct through first order from Hartree-Fock (HF) wave functions [l]. It is easily shown that errors in HF molecular geometry and barriers to internaf rotation are also of second order: Let the exact wave function J/ be expressed as * = (l*E2)-~(f$+EX), where r# is the HF wave function; +, I$, and x are normalized to unity; and @ and x are othrogonal. J.fA is a one electron operator and &I = CpIdlP), then because of Brillouin’s theorem @Jf Al x) - 0, and to lowest
order
A(A) = &>,
in E
- ti;jb 2 E2(&4& - (A)+).
(1)
Eq. (1) has been invoked to explain the good agreement between one electron properties which are obtained from experiment and HF calculations [l, 21. However, eq. (1) is of little practical utility since it gives no way of estimating E or (A),. Furthermore, ~2 is not necessarily small in many electron systems. However, in view of the ability to obtain good bond lengths, bond angles 133, and probably barriers to internal rotation [4] from HF theory, eq. (1) is extended to cover these properties. Because + and J/satisfy the Hellmann-Feynman theorem [5,6], &A(H) where R is a parameter
= A($ occurring
in the elec-
* NATO postdoctoral fellow. Present address: Department of Chemistry and James Franck Institute, University of Chicago, Chicago* Illinois, USA.
tronic cule, tronic brium
Hamiltonian H[?f For a diatomic moIeif R is taken to be the bond length, the elecenergy can be expanded about the equiLivalue
E=Eo+#(R-R,i2+
$ks(R-Re)3
t ..”
with a similar expansion for the HF energy. primes denote HF quantities,
, (2)
If
k
(31 (The term in k7 which $s neglected is of order g4.) The force (c?H. SRjXI Re can be estimated very
crudely as that of a “typical” HF excited s&.&e, and thus R;? - A, is correct through order E. Vnfortunately, because ((??H/%}2~ is singular, recent error bound techniques [2] carmot be used to estimate RL - Re. We can obtain a result analogous to eq. (3) for polyatomic molecules, where R represents a bond length or angle, As above we can neglect cubic and higher terms. In order to use eq. (3‘: for polyatomic molecufes, R, is the HF equilibrium value when all other coordinates are taken at their true equilibrium values. If the HF energy is expressed in terms of the HF geometry, we can either neglect the interaction force constants as being small, or interpret eq. (3) as a matrix equation which yields the same qualitative result. For an internal rotor a hose potential energy is adequately approximate’1 by the truncated potential 2 C V;2j COsR.j@, j=o
255
Volume 2, number 4
the barrier
CHEMICAL PHYSICS LETTERS
is equal to 2 Vn and [ 81
REFERENCES
and HF gives the barrier
correct through the first order. It should be noted that for any wave function (double primes) which satisfies the HellmannFeynman theorem,
(retaining only the quadratic part of the energy), and if an error bound could be calculated for a force (or torque operator), an experimental vailue of R, would give a bound for k and vice veysa. Discussions with Professor W. Klemperer R G. Parr are gratefully acknowledged.
256
August 1968
and
[l] J. Goodisman and W.Klemperer, J. Chem. Phys. 38 (1963) 721. [Z] P. Jennings and E. B. Wilson Jr., J. Chem. Phys. 47 (1967) 2130. [3] For instance: R. J. Buenker and S. D. Peyerimhoff. J. Chem. Phys. 45 (1966) 3682: S. D. Peyerimhoff, J. Chem. Phys. 47 (1967) 349. [4] For instance: W-H. Fink, D. C. Pan and L.C. Allen, J. Chem. Phys. 47 (1967) 895. [5] H. Hellmann. EinfUhrung in die Quantumchemie (F. Deuticke, Leipzig, 1937): R.P. Feynman, Phys. Rev. 56 (1939) 340. [S] R. E-Stanton, J. Chem.Phys. 36 (1962) 1298. [7J M. E.Schwartz, J. Chcm. Phys. 45 (1966) 4754. [8] J. Goodisman, J. Chem. Phys. 44 (1966) 2085: 45 (1966) 4689: 47 (1967) 334.