Zero-order Hamiltonian terms, derivative

In electronic structure calculations, it is not unlikely for a basis set to be dependent on the parameters. The most obvious case involves geometric parameters. The atomic orbital basis functions used to construct molecular orbitals are generally chosen to follow the atomic centers. This means that the functions are dependent on the molecular geometry, and so there will be nonzero derivatives of the usual one- and two-electron integrals. In the case of parameters such as an electric field strength, there is no functional dependence of the standard types of basis functions. The derivatives of all the basis functions with respect to this parameter are zero, and so all derivative integrals involving the zero-order Hamiltonian terms are zero as well. 

Comparing zero-order operators of Eqs. (2) and (3) one may observe that an advantage of Schmidt-orthogonalization is getting the zero-order Hamiltonian symmetric at least in the one-dimensional reference space spanned by 0). Left- and right-hand zero-order eigenvectors expressed in terms of determinants HF), and MR function 0) are listed in Table 1 for completeness. Detailed derivation of the reciprocal vectors has been shown in an earlier report [23],... 

Thus if we are only interested in deriving an effective Hamiltonian up to order X3, we need not concern ourselves with the explicit forms of S2 and S2. Furthermore, for the particular situation where X only has matrix elements which are off-diagonal in i] (i.e. all diagonal matrix elements of X are zero), the second term in (7.64) also vanishes and we have... 

If the zero-order wavefunction does, in fact, satisfy Eqn. (26), then the first term on the right in Eqn. (32) is identically zero. This means that the first derivatives are obtained as an expectation value of the derivative Hamiltonian. This last statement is the Hellmann-Feynman theorem. 

In the first subsection, we define the model function and various projection operators. For the case of a single-reference function, the term model function is synonymous with zero-order function . This is followed by a derivative of the effective Hamiltonian operator, and then definitions of the wave operator and the reaction operator. 

In the harmonic approximation only the term n = 1 is different from zero. It is then the anharmonicity that allows the coupling between the initial state and final states n > 1. The second-order derivative of the Hamiltonian does not pose any particular problem and can be evaluated in the same way as the first derivative (see above). 

Here by is the Kronecker delta, AX runs over the JT active coordinates, and i and j cover the set of AX components. The parameter Ka represents the harmonic force constant of the A mode. M is the electronic hamiltonian. The matrix element containing the derivative of versus Qax must be evaluated at the origin of coordinate space, as indicated by the zero subscript. It must be realized that the expression in Eq. (14) only contains the most essential contributions, viz. the harmonic terms and the linear terms. Further refinements include other quadratic interactions, such as the bilinear terms in Qm Qat> or ven higher order terms. 

The solution for the perturbed system can now be developed. The variable X is introduced as a scalar quantity that acts as a tunable dial for the perturbation in the range of 0 S 1. When X is equal to zero, there is no perturbation resulting in the unperturbed system. When the value of X is unity, the system experiences the full perturbation. At the end of the derivation, the value of X will be set at unity removing it from all of the expressions and the perturbation will be entirely reflected in the first and higher order perturbing Hamiltonians. The Hamiltonian for the perturbed system can be written as an expansion series in terms of X. 

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