Hessian matrices Hamiltonian

In such a case the last choice is to take the direction of the eigenvector of the only one nonzero eigenvalue of the rank one Hessian matrix of the difference between the two adiabatic potential energies [51]. In the vicinity of conical intersection, the topology of the potential energy surface can be described by the diadiabatic Hamiltonian in the form... 

This equation determines a rank-1 matrix, and the eigenvector of its only one nonzero eigenvalue gives the direction dictated by the nonadiabatic couphng vector. In the general case, the Hamiltonian differs from Eq.(l), and the Hessian matrix has the form... 

Let us finally take a closer look at the orbital Hessian matrix, H(00). The calculation now involves the evaluation of commutators between the Hamiltonian and products of excitation operators according to equation (4 14). In spite of the rather tiring algebra, the result takes a surprisingly simple form ... 

Since q, p ) is the degree of freedom corresponding to the negative eigenvalue of the Hessian matrix of the potential, the sign of its lowest potential term is minus—that is, the inverse harmonic potential. For other degrees of freedom, the lowest-order terms of their potentials describe harmonic oscillators. Therefore, they are vibrational degrees of freedom. In Eq. (16), we normalize the coefficients of the Hamiltonian so that they are written cls ( m = 1,2,..., N). 

We now examine the elements of the gradient vector and Hessian matrix in more detail. The Hamiltonian operator of Eq. (115) and the generator commutation relations of Eqs (120) and (121) result in the following operator... 

Finally the state Hessian matrix M is seen from Eq. (149) to be proportional to the representation of the Hamiltonian operator in the orthogonal complement basis, but with all the eigenvalues shifted by the constant amount (0). The dimension of the matrix M will be one less than the length of the CSF expansion unless it is constructed in the linearly dependent projected basis or the overcomplete CSF expansion set basis. Since the Hamiltonian matrix must usually be constructed in the CSF basis in the MCSCF method anyway, it is most convenient if M and C are also constructed in this basis. The transformation to the projected basis, if explicitly required, involves the projection matrix (1 — cc ). The matrix M only requires the two-electron integral subset that consists of all four orbital indices corresponding to occupied orbitals. 

Next consider two rotation operators, T and T ,., that satisfy Eq. (242). Substitution of Eq. (242) into the expression for the blocks of the Hessian matrix, under the assumption that the current CSF expansion coefficients are an eigenvector of the current Hamiltonian matrix, gives... 

In these equations it is also assumed that the orthogonal complement states have been chosen to diagonalize the Hamiltonian matrix. This is merely a formal convenience and is not necessary for the validity of the results of this section. Substitution of these identities into the expression for the partitioned orbital Hessian matrix gives... 

Hamiltonian With these operators the two off-diagonal blocks E and E of the electronic Hessian matrix and S and of the overlap matrix become zero [see Exercise 3.12] and the propagator in Eq. (3.159) can be written as... 

The formulas above give the gradient and the Hessian in terms of matrix elements of the excitation operators. They can be evaluated in terms of one-and two-electron integrals, and first and second order reduced density matrices, by inserting the Hamiltonian (3 24) into equations (4 9), (4 11), and (4 13)-(4 15). Note that transition density matrices and are needed for the evaluation of the Cl coupling matrix (4 15). 

E n if one starts from SCF estimates of a complex nature, only real SCF Wave functions have been obtained as a result of the calculations, and this means that both the occupied and virtual SCF orbitals form a real orthonoimalized set, spanning the whole subspace of the LCAO basis orbitals, as noted previously. This implies also that all the matrix elements of ttie Hamiltonian between determlnantal wave functions built up from these SCF orbitals are necessarily real. UtllMng this fact, and following the derivation given in [20], the Hessian can be given as... 

All matrix elements in the Newton-Raphson methods may be constructed from the one- and two-particle density matrices and transition density matrices. The linear equation solutions may be found using either direct methods or iterative methods. For large CSF expansions, such micro-iterative procedures may be used to advantage. If a micro-iterative procedure is chosen that requires only matrix-vector products to be formed, expansion-vector-dependent effective Hamiltonian operators and transition density matrices may be constructed for the efficient computation of these products. Sufficient information is included in the Newton-Raphson optimization procedures, through the gradient and Hessian elements, to ensure second-order convergence in some neighborhood of the final solution. 

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