2.5) As we know, we do not have exact equations for the molecular orbitals |øn); instead, they are represented as a linear combination of atomic orbitals. For H, in the minimal basis, for example, lø1) 1 (|1sA) + |18B)). V 2(1 + S) In general, we have: M |øn) = Cmn |Xm), т where |Xm) is an atomic orbital, and there are M total AOs in the basis. In the LCAO basis, the Hartree-Fock equation can be written in matrix form by the Roothaan-Hall equation: FC = SCe. In this equation, F is the Fock matrix expressed in the AO basis, C is the matrix of AO expansion coefficients, S is the matrix of overlap integrals for the basis, and e are the Hartree-Fock orbital energies (a diagonal matrix). The SCF procedure involves iteratively solving this equation. Each element of the Fock matrix F for a closed-shell, N-electron molecule is given by: N/2 M,M 1 1 Fij Hgre +EE CraCia ( 2( xłxa x}x3)- (xix3 r12 r12 a 1,0 HCore is the independent particle Hamiltonian matrix (called the core Hamiltonian). Use this equation to write out the F12 matrix element for H, in the minimal AO basis, using the solutions to the independent particle problem as the coefficients in the C matrix (this is typically how the Fock matrix is generated in the first iteration of the SCF procedure).
2.5) As we know, we do not have exact equations for the molecular orbitals |øn); instead, they are represented as a linear combination of atomic orbitals. For H, in the minimal basis, for example, lø1) 1 (|1sA) + |18B)). V 2(1 + S) In general, we have: M |øn) = Cmn |Xm), т where |Xm) is an atomic orbital, and there are M total AOs in the basis. In the LCAO basis, the Hartree-Fock equation can be written in matrix form by the Roothaan-Hall equation: FC = SCe. In this equation, F is the Fock matrix expressed in the AO basis, C is the matrix of AO expansion coefficients, S is the matrix of overlap integrals for the basis, and e are the Hartree-Fock orbital energies (a diagonal matrix). The SCF procedure involves iteratively solving this equation. Each element of the Fock matrix F for a closed-shell, N-electron molecule is given by: N/2 M,M 1 1 Fij Hgre +EE CraCia ( 2( xłxa x}x3)- (xix3 r12 r12 a 1,0 HCore is the independent particle Hamiltonian matrix (called the core Hamiltonian). Use this equation to write out the F12 matrix element for H, in the minimal AO basis, using the solutions to the independent particle problem as the coefficients in the C matrix (this is typically how the Fock matrix is generated in the first iteration of the SCF procedure).
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Transcribed Image Text:2.5) As we know, we do not have exact equations for the molecular orbitals |øn); instead, they are
represented as a linear combination of atomic orbitals. For H, in the minimal basis, for example,
1
(|18A) + |1sB)).
2(1+ S)
In general, we have:
M
|øn) = Cmn |Xm),
m
where Xm) is an atomic orbital, and there are M total AOs in the basis. In the LCAO basis, the
Hartree-Fock equation can be written in matrix form by the Roothaan-Hall equation:
FC = SCe.
In this equation, F is the Fock matrix expressed in the AO basis, C is the matrix of AO expansion
coefficients, S is the matrix of overlap integrals for the basis, and e are the Hartree-Fock orbital
energies (a diagonal matrix). The SCF procedure involves iteratively solving this equation.
Each element of the Fock matrix F for a closed-shell, N-electron molecule is given by:
N/2 M,M
1
1
Fij = Hre
Hgre +EECraC%a ( 2 ( xix
xix3 )).
rcore
r12
r12
1,0
a
HCore is the independent particle Hamiltonian matrix (called the core Hamiltonian). Use this
equation to write out the F12 matrix element for H, in the minimal AO basis, using the solutions
to the independent particle problem as the coefficients in the C matrix (this is typically how the
Fock matrix is generated in the first iteration of the SCF procedure).
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