One-index transformations

Table 4. The isotropic indirect spin-spin coupling constant of calculated at various levels of theory. LL refers to the Levy-Leblond Hamiltonian, std refers to a full relativistic calculation using restricted (RKB) or unrestricted (UKB) kinetic balance, spf refers to calculations based on a spin-free relativistic Hamiltonian. Columns F, G and whether quaternion imaginary parts are deleted (0) or not (1) from the regular Fock matrix F prior to one-index transformation, from the two-electron Fock matrix G...

Direct evaluation of this four-fold sum would require 4didi dgdpjdgdftd c multiplications to form a symmetry block of R integrals. By constrast, sequential one-index transformations... 

We define a new set of operators using one-index transformed integrals [15]... 

All formulae until now are independent of the spin character of the perturbation. The spin complications enter at the level of one-index transformations. If k has been derived from an operator with spin, i.e. if it is given by Eq. (33) with the minus sign, the one-index transformation will change the spin properties of the Hamiltonian. In... 

A linear transformation of a configuration vector cb thus requires the construction of a configuration gradient with B> as the reference state [Eq. (94)], and the construction of an orbital gradient with a symmetric transition density matrix [Eq. (95)]. A linear transformation on an orbital vector °b requires the construction of a configuration gradient [Eq. (96)] and an orbital gradient [Eq. (97)] from the one-index transformed Hamiltonian K. 

After omitting all contributions to the MCSCF derivatives which arise from variations in the configuration space, the remaining terms either already appear in the AO basis (e.g., all terms index transformed integrals.) This implies that SCF derivatives can be calculated completely in the AO basis. We give some examples to make this point clear. 

Consider first the SCF response equations [Eq. (64)]. The construction of /differentiated integrals and from one-index transformed integrals. Both matrices may be calculated in the AO basis (see Appendix E). Furthermore, the solution of the response equations requires linear transformations of orbital trial vectors. Equation... 

The evaluation of F(n) requires the construction of Fock matrices from multiply one-index transformed integrals. For example, F(3) contains the term... 

The explicit calculation of C<0)/hf s avoided by modifying the densities Deff and deff by one-index transformed HF densities — DHF and... 

The term T2 [Eq. (252)] may be calculated in a variety of ways, noting that all contributions to g l are already available in the MO basis. One may either carry out the one-index transformations according to Eq. (249) and then combine glU with the densities t( ]b. Alternatively, one may construct intermediate MP2 Fock matrices from differentiated and one-index transformed integrals and calculate T2 according to... 

In these expressions differentiation and one-index transformations refer to the g integrals only of the Fock matrix [Eqs. (235) and (236)], treating the t elements as densities. The Fock matrix density elements in (Dj 1 and to the contravariant representation. If first derivative integrals in the MO basis is reduced to two occupied and two unoccupied indices (Handy et al., 1986). Note that 7] + T2 [Eqs. (257) and (258)] has the same structure as the <2) part of the MRCI Hessian (129). 

One-index transformations with A as the transformation matrix are defined as... 

In the MCSCF case the undifferentiated Fock matrix is symmetric since the orbital optimization ensures that ffj = 2(FfJ — FfJ) = 0. The Fock matrix also appears in the calculation of expectation values of one-index transformed Hamiltonians (see Appendix F). 

The Fock matrix from one-index transformed integrals is... 

In the MCSCF case this matrix may be calculated according to Eqs. (E.3)-(E.5), but using matrices Fl" aFln and Qln] containing one-index transformed integrals. For example, for the inactive Fock matrix we obtain... 

The Fock matrix with doubly one-index transformed integrals may be calculated in the same way. In particular, the inactive and active Fock matrices... 

The first term corresponds to a doubly one-index transformed inactive Fock matrix. The second term is a one-index transformed matrix P(B)1"1. The third and fourth terms may be constructed in the AO basis with the matrices multiplying the [Pg.240]

Appendix F. Expectation Values of One-Index Transformed Hamiltonians... 

The expectation value of the one-index transformed Hamiltonian can be written in the following alternative ways (Helgaker, 1986)... 

In the first expression the integrals are in the covariant AO representation (in which they are calculated), and the one-index transformed density elements are in the contravariant representation (obtained from the MO basis in usual one- and two-electron transformations). The second expression is useful whenever the transformation matrix is calculated directly in the covariant AO representation and requires the transformation of the Fock matrix to the contravariant representation. The last expression is convenient when the number of perturbations is large, since it avoids the transformation of the covariant AO Fock matrix to the MO or contravariant AO representations. 

Furthermore, we have defined the one-index transformed integrals as... 

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