First-derivative coupling matrix equation

Born-Huang expansion, 286—289 first-derivative coupling matrix, 290—291 nuclear motion Schrodinger equation, 289-290... 

The ADT matrix for the lowest two electronic states of H3 has recently been obtained [55]. These states display a conical intersection at equilateral triangle geometi ies, but the GP effect can be easily built into the treatment of the reactive scattering equations. Since, for two electronic states, there is only one nonzero first-derivative coupling vector, w5 2 (Rl), we will refer to it in the rest of this... 

To evaluate a given row of the A,B oxC matrix it can readily be seen what has to be done from the equation set. The partial derivatives of one of the functions must be evaluated with respect to each solution variable, each variable first derivative and each variable second derivative. These partial derivatives allow the elements in Eq. (11.60) to be evaluated row by row, or column by column if more convenient. In fact it is more efficient to evaluate the terms on a column by column basis as one then only has to select an increment value once for each variable and derivative and then apply this incremented value to all the functions. In terms of a single variable BV problem, there is roughly as much computational effort in setting up the finite difference equations, where N is the number of coupled equations. Then there is additional time required to solve the set of coupled matrix equations as represented by Eq. (11.58). 

The first derivative of the density matrix with respect to the magnetic induction (dPfiv/dBi) is obtained by solving the coupled-perturbed Hartree-Fock (or Kohn-Sham) equations to which the first derivative of the effective Fock (or Kohn-Sham) operator with respect to the magnetic induction contributes. Due to the use of GIAOs, specific corrections arising from the effective operator Hcnv describing the environment effects will appear. We refer to Ref. [28] for the PCM model and to Ref. [29] for the DPM within either a HF or DFT description of the solute molecule. 

The magnetic hyperfine interaction terms were given in equation (8.351) and the electric quadrupole interaction in equation (8.352). We extend the basis functions by inclusion of the 7Li nuclear spin I, coupled to J to form F the value of / is 3/2. We deal with each term in turn, first deriving expressions for the matrix elements in the primitive basis set (8.353), and then extending these results to the parity-conserved basis. All matrix elements are diagonal in F, and any elements off-diagonal in S and / can of course be ignored. 

On the contrary, the density derivatives are necessary to get higher order free energy derivatives in particular the second free energy derivatives require the first derivative of the density matrix P , which can be obtained by solving an appropriate coupled perturbed Hartree-Fock (CPHF) or Kohn-Sham (CPKS) equations. 

Once again, the derivative of the density matrix P can be obtained as solution of first-order coupled-perturbed Hartree-Fock equation with derivar tive Fock matrix given by eq. (1.70), exactly as for the nuclear shielding. 

In the coupled perturbed Kohn-Sham method, the first wavefunction derivatives are given by calculating the first derivatives of the orbitals in terms of perturbations. The Kohn-Sham method is based on the Slater determinant. Therefore, since the Kohn-Sham wavefunction is represented with orbitals, the corresponding first wavefunction derivatives are also described by the first derivatives of the orbitals. For simplicity, let us consider the Kohn-Sham-Roothaan equation in Eq. (4.13), which is a matrix equation using basis functions based on the Roothaan method. 

Using Eqs. (4.61) and (4.63), matrix U is calculated to give the response properties in terms of the uniform electric field dipole moments, polarizabilities, hyperpolarizabilities, and so forth. Equation (4.61) is called the coupled perturbed Kohn-Sham equation. Other response properties are calculated by solving Eq. (4.61) after setting the first derivative of the Fock operator, F, in terms of each perturbation. Note, however, that this method has problems in actual calculations similarly to the time-dependent response Kohn-Sham method. For example, using most functionals, this method tends to overestimate the electric field response properties of long-chain polyenes. 

Note that we obtain two equations of the same type as Eq. (9.248) for our coupled system of differential equations, which can be combined into a single hepta-diagonal matrix equation that further allows us to determine the two radial functions simultaneously To derive this hepta-diagonal system we apply Eq. (G.25) (see appendix G) to the transformed coupled Eqs. (9.227) and (9.228) and arrange all 2 x n discretized equations into a matrix equation. The first... 

Besides the evaluation of the Fock operator, density matrix, and integral derivatives, the analytical evaluation of energy second derivatives is achieved once the density matrix first derivatives with respect to geometric perturbations are known. The evaluation of the latter terms is the bottleneck of computational procedures because of its cost in terms of CPU time and disk storage. In the common practice, such a quantity is obtained by resorting to the first-order coupled perturbed HF (or KS) technique [4], which conceptually starts from the HF (or KS) equations, expands all the matrices in terms of the perturbation, and, by collecting all the terms at the same order, yields sets of equations which are usually solved iteratively. 

The Jacobian matrix A can be shown [see Exercise 11.9) to be the first derivative of the time-independent coupled cluster amphtude equations, i.e. the coupled cluster vector function e, Eq. (9.81), with respect to the time-independent amplitudes... 

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