Wave function expansions coefficient

So far, we focused on conventional quantum chemical approaches that approximate the FCI wave function by truncating the complete N-particle Hilbert space based on predefined configuration selection procedures. In a different approach, the number of independent Cf coefficients can be reduced without pruning the FCI space. This is equivalent to seeking a more efficient parameterization of the wave function expansion, where the Cl coeflBcients are approximated by a smaller set of variational parameters that allow for an optimal representation of the quantum state of interest. Different approaches, which we will call modern solely to distinguish them from the standard quantum chemical methods, have emerged from solid-state physics. 

From Eq. (8.14), we can immediately read off the Cl coefficients of the wave function expansion. If we write the summation over the elements of the eigenvector of Hgi explicitly, we obtain for the Cl coefficient of one Slater determinant n... 

We have derived the Hartree-Fock method from the requirement that the orbitals should be those that give the lowest energy expectation value of a Slater determinant. In a Cl expansion where only singly substituted Slater determinants are used, all coefficients for the latter are equal to zero according to Brillouin s theorem. However, if doubly substituted Slater determinants (and higher) are introduced in the total wave function, the coefficients for the singly substituted Slater determinants are no longer equal to zero. 

The effect on the energy and wave function of such occurrences is known as nondynamical correlation (also known as near-degeneracy correlation). Apart from the obvious criterion of the size of the largest coefficients in the wave function expansion beside Lee and Taylor proposed a simple diagnostic known as the T diagnostic... 

The time dependence of the molecular wave function is carried by the wave function parameters, which assume the role of dynamical variables [19,20]. Therefore the choice of parameterization of the wave functions for electronic and nuclear degrees of freedom becomes important. Parameter sets that exhibit continuity and nonredundancy are sought and in this connection the theory of generalized coherent states has proven useful [21]. Typical parameters include molecular orbital coefficients, expansion coefficients of a multiconfigurational wave function, and average nuclear positions and momenta. We write... 

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

The gradient of the PES (force) can in principle be calculated by finite difference methods. This is, however, extremely inefficient, requiring many evaluations of the wave function. Gradient methods in quantum chemistiy are fortunately now very advanced, and analytic gradients are available for a wide variety of ab initio methods [123-127]. Note that if the wave function depends on a set of parameters X], for example, the expansion coefficients of the basis functions used to build the orbitals in molecular orbital (MO) theory. 

The problem has now become how to solve for the set of molecular orbital expansion coefficients, c. . Hartree-Fock theory takes advantage of the variational principle, which says that for the ground state of any antisymmetric normalized function of the electronic coordinates, which we will denote H, then the expectation value for the energy corresponding to E will always be greater than the energy for the exact wave function ... 

Without introducing any approximations, the total (exact) wave function can be written as an expansion in the complete set of electronic functions, with the expansion coefficients being functions of the nuclear coordinates. 

This is perhaps the easiest method to understand. It is based on the variational principle (Appendix B), analogous to the HF method. The trial wave function is written as a linear combination of determinants with the expansion coefficients determined by requiring that the energy should be a minimum (or at least stationary), a procedure known as Configuration Interaction (Cl). The MOs used for building the excited Slater determinants are taken from a Hartree-Fock calculation and held fixed. Subscripts S, D, T etc. indicate determinants which are singly, doubly, triply etc. excited relative to the... 

The expansion coefficients determine the first-order correction to the perturbed wave function (eq. (4.35)), and they can be calculated for the known unperturbed wave functions and energies. The coefficient in front of 4>o for 4 i cannot be determined from the above formula, but the assumption of intermediate normalization (eq. (4.30)) makes Co = 0. 

Bernal, M. J. M., and Boys, S. F., Trans. Roy. Soc. [London) A245, 116, (i) Electronic wave functions. VII. Methods of evalua-ating the fundamental coefficients for the expansion of vector-coupled Schrodinger integrals and some values of these. ... 

The first term on the left-hand side of equation (10.18) has the form of a Schrodinger equation for nuclear motion, so that we may identify the expansion coefficient Xk Q) as a nuclear wave function for the electronic state k. The second term couples the influence of all the other electronic states to the nuclear motion for a molecule in the electronic state k. 

---