Linear variation method hamiltonian

Variational methods, in particular the linear variational method, are the most widely used approximation techniques in quantum chemistry. To implement such a method one needs to know the Hamiltonian H whose energy levels are sought and one needs to construct a trial wavefunction in which some flexibility exists (e.g., as in the linear variational method where the Cj coefficients can be varied). In Section 6 this tool will be used to develop several of the most commonly used and powerful molecular orbital methods in chemistry. 

Among the classes of the trial wave functions, those employing the form of the linear combination of the functions taken from some predefined basis set lead to the most powerful technique known as the linear variational method. It is constructed as follows. First a set of M normalized functions dy, each satisfying the boundary conditions of the problem, is selected. The functions dy are called the basis functions of the problem. They must be chosen to be linearly independent. However we do not assume that the set of fdy is complete so that any T can be exactly represented as an expansion over it (in contrast with exact expansion eq. (1.36)) neither is it assumed that the functions of the basis set are orthogonal. A priori they do not have any relation to the Hamiltonian under study - only boundary conditions must be fulfilled. Then the trial wave function (D is taken as a linear combination of the basis functions dyy... 

Thus the basis of Slater determinants can be used as a basis in a linear variational method eq. (1.42) when the Hamiltonian dependent or acting on coordinates of N electrons is to be studied. The problem with this theorem is that for most known choices of the basis of spin-orbitals used for constructing the Slater determinants of eq. (1.137) the series in eq. (1.138) is very slow convergent. We shall address this problem later. 

The above classification of asymmetric potential functions is convenient for comparison of different molecules or as a systematic basis for making an initial fit to experimental data. However, when the Schrodinger equation is being solved by the linear variation method with harmonic-oscillator basis functions, it may not provide the best choice of origin for the basis function. For example, a better choice in the case of an asymmetric double-minimum oscillator, where accurate solutions are required in both wells, would be somewhere between the two wells. Systematic variation of the parameters may still be made as outlined above, but the origin should be translated before the Hamiltonian matrix is set up. The equations given earlier... 

Thus the energy of this wave function is below the HF energy by the pair correlation energy. To obtain the best possible energy for the above pair-function, we use the linear variation method. Thus we construct the matrix representation of the Hamiltonian in the subspace spanned by q> and all double excitations involving Xa Xb lowest eigenvalue... 

Having a hamiltonian and a trial wavefunction, we are now in a position to use the linear variation method. The detailed derivation of the resulting equations is complicated and notationally clumsy, and it has been relegated to Appendix 7. Here we discuss the results of the derivation. 

The lack of correlation is the actual source of aU errors. In particular, a Slater determinant incorporates exchange correlation, i.e., the motion of two electrons with parallel spins is correlated (the so-called Fermi correlation). Unfortunately, the motion of electrons with opposite spins remains uncorrelated. It is common to define correlation energy, corr> as the difference between the exact nonrelativistic energy of the system, eo, and the Hartree-Fock energy, Eo, obtained in the limit that the basis set approaches completeness Ecorr = fo - o- The simplest manner to understand the inclusion of the correlation effects is through the method of configuration interaction (Cl). The basic idea is to diagonalize the N-electron Hamiltonian in a basis of N-electron functions we represent the exact wave function as a linear combination of N-electron trial functions and use the linear variational method. 

For present purposes it is more useful to concentrate on other approaches, which start from the finite-basis form of the linear variation method. In many forms of variation-perturbation theory, exact unperturbed eigenfunctions are not required and the partitioning of the Hamiltonian into two terms is secondary to a partitioning of the basis. At the same time, as we shall see, it is possible to retrieve the equations of conventional perturbation theory by making an appropriate choice of basis. 

In the linear variation method, we can obtain conditions that ensure symmetry-adapted solutions even in the absence of symmetry-adapted basis vectors. Consider some syrmnetry associated with the operator A that commutes with the exact Hamiltonian... 

The simple and extended Hiickel methods are not rigorous variational calculations. Although they both make use of the secular determinant technique from linear variation theory, no hamiltonian operators are ever written out explicitly and the integrations in Hij are not performed. These are semiempirical methods because they combine the theoretical form with parameters fitted from experimental data. 

The ab initio SCF cluster wavefunction has been used to investigate the bonding of CO and CN- on Cu,0 (5,4,1), (5 surface layer, 4 second layer and 1 bottom layer atoms), and to calculate their field dependent vibrational frequency shifts in fields up to 5.2 x 107 V/cm(46). A schematic view of the Cu10 (5,4,l)CO cluster is shown in Figure 8. In order to assess the significance of Lambert s proposal, that the linear Stark effect is the dominant factor in the field dependent frequency shift, the effect of the field was calculated by three methods. One is by a fully variational approach (i.e., the adsorbate is allowed to relax under the influence of the applied field) in which the Hamiltonian for the cluster in a uniform electric field, F, is given by... 

Kohn-variational (12), Schwinger-variational, (13) R-Matrix (14), and linear algebraic techniques (15,16) have been quite successful in calculating collisional and phH oTo nization cross sections in both resonant and nonresonant processes. These approaches have the advantage of generality at the cost of an explicit treatment of the continuous spectrum of the Hamiltonian and the requisite boundary conditions. In the early molecular applications of these scattering methods, a rather direct approach based on the atomic collision problem was utilized which lacked in efficiency. However in recent years important conceptual and numerical advances in the solution of the molecular continuum equations have been discovered which have made these approaches far more powerful than those of a decade ago... 

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