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4 From Orbital Models to Accurate Predictions
4.7 Make a rough estimate of the total number of determinants in the MRCISD wave function for a system with 74 electrons, 154 orbitals and a
CAS(2,2)CI reference wave function. Calculate the percentage of 2h-2p excitations in the MR-CISD wave function (neglect the 1h,1p,1h-1p,2h and 2p
excitations, they give rise to a very small number of determinants).
The justification for eliminating the 2h-2p determinants relies on second-order
perturbation theory in its quasi-degenerate formulation as exposed in Chap. 1.
Although it can be done for an arbitrary number of unpaired electrons, we will
elaborate the 2-electrons/2-orbitals case for simplicity. The model space is spanned
by the neutral and ionic determinants
Φ I ={|hhab|, |hhba|, |hhaa|, |hhbb|}
(4.31)
where h is one of the inactive orbitals, doubly occupied in all determinants of the
model space. The lowest two eigenstates of the model space are the singlet and triplet
spin functions whose energy difference is related to J. However, before diagonalizing
we will first evaluate the effect of the 2h-2p external determinants on the matrix
elements between the determinants of the model space with QDPT. First, we take
a look at the off-diagonal elements and calculate the second-order contributions of
Φ R =| ppab| and Φ S =| ppba| to the dressed matrix element of Φ I =| hhab| and
Φ J =| hhba| according to the expression given in Eq. 1.86.The2h-2p determinant
Φ R is obtained by making a double replacement in Φ I , exciting the electrons in
h to the unoccupied orbitals p and Φ S arises from Φ J in an analogous way. The
contributions to the effective matrix element are
Φ R :
hhab| ˆ
V |ppabppab| ˆ
V |hhba
E J − E R
(4.32a)
Φ S :
hhab| ˆ
V |ppbappba| ˆ
V |hhba
E J − E S
(4.32b)
For the contribution of Φ R , the second matrix element in the numerator is zero
because the determinants on the left and the right of the operator have more than
two different columns, and the same occurs for the first matrix element in the Φ S
contribution. This eliminates any second-order perturbation contribution from the
2h-2p determinants to the off-diagonal elements of the model space.
4.8 Write down the second-order contribution of Φ Q to Φ I | ˆ
H eff |Φ L , where
Φ Q arises from a double excitation from orbital h to orbital p a c t i n go nt h e
ionic determinant Φ L =|hhbb|. Argue that this contribution is equal to zero.
4 From Orbital Models to Accurate Predictions
4.7 Make a rough estimate of the total number of determinants in the MRCISD wave function for a system with 74 electrons, 154 orbitals and a
CAS(2,2)CI reference wave function. Calculate the percentage of 2h-2p excitations in the MR-CISD wave function (neglect the 1h,1p,1h-1p,2h and 2p
excitations, they give rise to a very small number of determinants).
The justification for eliminating the 2h-2p determinants relies on second-order
perturbation theory in its quasi-degenerate formulation as exposed in Chap. 1.
Although it can be done for an arbitrary number of unpaired electrons, we will
elaborate the 2-electrons/2-orbitals case for simplicity. The model space is spanned
by the neutral and ionic determinants
Φ I ={|hhab|, |hhba|, |hhaa|, |hhbb|}
(4.31)
where h is one of the inactive orbitals, doubly occupied in all determinants of the
model space. The lowest two eigenstates of the model space are the singlet and triplet
spin functions whose energy difference is related to J. However, before diagonalizing
we will first evaluate the effect of the 2h-2p external determinants on the matrix
elements between the determinants of the model space with QDPT. First, we take
a look at the off-diagonal elements and calculate the second-order contributions of
Φ R =| ppab| and Φ S =| ppba| to the dressed matrix element of Φ I =| hhab| and
Φ J =| hhba| according to the expression given in Eq. 1.86.The2h-2p determinant
Φ R is obtained by making a double replacement in Φ I , exciting the electrons in
h to the unoccupied orbitals p and Φ S arises from Φ J in an analogous way. The
contributions to the effective matrix element are
Φ R :
hhab| ˆ
V |ppabppab| ˆ
V |hhba
E J − E R
(4.32a)
Φ S :
hhab| ˆ
V |ppbappba| ˆ
V |hhba
E J − E S
(4.32b)
For the contribution of Φ R , the second matrix element in the numerator is zero
because the determinants on the left and the right of the operator have more than
two different columns, and the same occurs for the first matrix element in the Φ S
contribution. This eliminates any second-order perturbation contribution from the
2h-2p determinants to the off-diagonal elements of the model space.
4.8 Write down the second-order contribution of Φ Q to Φ I | ˆ
H eff |Φ L , where
Φ Q arises from a double excitation from orbital h to orbital p a c t i n go nt h e
ionic determinant Φ L =|hhbb|. Argue that this contribution is equal to zero.