Optimal wave function, dependence

Thus, as we shall see in Chapters 19, 20 and 28, having in mind that the angular parts of new wave functions depend on orbital quantum number / only in the form of a phase multiplier, we are in a position to obtain straightforwardly optimal expressions for all matrix elements needed. 

The MCSCF/CM response method provide procedures for obtaining frequency-dependent molecular properties when investigating a molecule coupled to a structured environment and the basis is achieved by treating the quantum mechanical subsystem on a quantum mechanical level and the structured environment as a classical subsystem described by a molecular mechanics force field. The important interactions between the two subsystems are included directly in the optimized wave function. 

Once the gas phase Hamiltonian is parametrized as a function of the inner-sphere reaetion coordinate(s), the free energy is calculated as a function of the proton coordinate(s), the scalar solvent coordinates, and the inner-sphere reaction coordinate(s). Note that this approaeh assumes that the optimized geometries of the VB states are not significantly affected by the solvent. For proton transfer reactions, the proton donor-acceptor distance may be treated as an additional solute reaction coordinate that ean be incorporated into the molecular mechanical terms describing the diagonal matrix elements hf- and, in some cases, the off-diagonal matrix elements (/io)y. If the inner-sphere reaction coordinate represents a slow mode, it is treated in the same way as the solvent coordinates. As discussed throughout the literature, however, often the inner-sphere reaction coordinate must be treated quantum mechanically [27, 28]. In this case, the inner-sphere reaction coordinate is treated in the same way as the proton coordinate(s), and the vibrational wave functions depend explicitly on both the proton coordinate(s) and the inner-sphere reaction coordinate(s). 

Before leaving minimal basis H2 (at least, temporarily), we want to use the model to illustrate exponent optimization. We have been using a standard exponent of C = 1.24. These orbital exponents are nonlinear parameters upon which our wave function depends. By the variational principle, the best wave... 

In the case of forces employing the mixed distribution most approaches neglect terms describing the dependence on the coefficients. In case of the mixed distribution this term is related to their influence on the nodal surface for iI/q and hard to estimate. Usually the contribution of this term is accepted to be small for an energy optimized wave function. One obtains... 

The above procedure must be repeated for self-consistency since the Fock operator itself depends on the optimized wave function. In this way, we have established a fixed-point sequence of iterations (where each iteration involves the diagonalization of the Fock matrix) which - if it converges - will lead us to the optimized Hartree-Fock wave function. Conveigence of the iterations is not guaranteed, however, but will depend on how we choose those elements of the Fock matrix that are not determined by the zero-order conditions (10.9.9). 

If the wave function is variationally optimized with respect to all parameters (HF or MCSCF, but not Cl), the last term disappears since the energy is stationary with respect to a variation of the MO/state coefficients (Ho,Pi and P2 do not depend on the parameters C). 

The first step beyond the statistical model was due to Hartree who derived a wave function for each electron in the average field of the nucleus and all other electrons. This field is continually updated by replacing the initial one-electron wave functions by improved functions as they become available. At each pass the wave functions are optimized by the variation method, until self-consistency is achieved. The angle-dependence of the resulting wave functions are assumed to be the same as for hydrogenic functions and only the radial function (u) needs to be calculated. 

Since the nuclei are massive compared to electrons, m/M << 1, they move so slowly in a molecule that the electrons can always maintain their optimal motion at each Q. The electronic wave functions may therefore be assumed to depend on the nuclear positions only and not on their momenta, i.e. Naip Q,q) — 0. The electrons are said to follow the nuclei adiabatically. Under these assumptions equation (30) reduces to... 

Figure 8. (a) Pulse sequence resulting from optimization of the control field to generate H in the same reaction as studied in Fig. 6. (6) The Husimi transform of the pulse sequence shown in (a). (c) Time dependence of the norms of the ground-state and excited-state populations as a result of application of the pulse sequence shown in (a). Absolute value of the ground-state wave function at 1500 au (37.5 fs) propagated under the pulse sequence shown in (a), shown superposed on a contour diagram of the ground-state potential energy surface. (From D. J. Tannor and Y. Jin, in Mode Selective Chemistry, B. Pullman, J. Jortner, and R. D. Levine, Eds. Kluwer, Dordrecht, 1991.)...

The formulation of the calculation of the optimal control field that guides the evolution of a quantum many-body system relies, basically, on the solution of the time-dependent Schrodinger equation. Messina et al. [25] have proposed an implementation of the calculation of the optimal control field for an n-degree-of-freedom system in which the Hartree approximation is used to solve the time-dependent Schrodinger equation. In this approximation, the n-degree-of-freedom wave function is written as a product of n single-degree-of-freedom wave functions, and the factorization is assumed to be valid for all time. 

We shall in this chapter discuss the methods employed for the optimization of the variational parameters of the MCSCF wave function. Many different methods have been used for this optimization. They are usually divided into two different classes, depending on the rate of convergence first or second order methods. First order methods are based solely on the calculation of the energy and its first derivative (in one form or another) with respect to the variational parameters. Second order methods are based upon an expansion of the energy to second order (first and second derivatives). Third or even higher order methods can be obtained by including more terms in the expansion, but they have been of rather small practical importance. 

As was mentioned in Chapter 2, there exists another method of constructing the theory of many-electron systems in jj coupling, alternative to the one discussed above. It is based on the exploitation of non-relativistic or relativistic wave functions, expressed in terms of generalized spherical functions [28] (see Eqs. (2.15) and (2.18)). Spin-angular parts of all operators may also be expressed in terms of these functions (2.19). The dependence of the spin-angular part of the wave function (2.18) on orbital quantum number is contained only in the form of a phase multiplier, therefore this method allows us to obtain directly optimal expressions for the matrix elements of any operator. The coefficients of their radial integrals will not depend, except phase multipliers, on these quantum numbers. This is the case for both relativistic and non-relativistic approaches in jj coupling. 

Continuum solvation models consider the solvent as a homogeneous, isotropic, linear dielectric medium [104], The solute is considered to occupy a cavity in this medium. The ability of a bulk dielectric medium to be polarized and hence to exert an electric field back on the solute (this field is called the reaction field) is determined by the dielectric constant. The dielectric constant depends on the frequency of the applied field, and for equilibrium solvation we use the static dielectric constant that corresponds to a slowly changing field. In order to obtain accurate results, the solute charge distribution should be optimized in the presence of the field (the reaction field) exerted back on the solute by the dielectric medium. This is usually done by a quantum mechanical molecular orbital calculation called a self-consistent reaction field (SCRF) calculation, which is iterative since the reaction field depends on the distortion of the solute wave function and vice versa. While the assumption of linear homogeneous response is adequate for the solvent molecules at distant positions, it is a poor representation for the solute-solvent interaction in the first solvation shell. In this case, the solute sees the atomic-scale charge distribution of the solvent molecules and polarizes nonlinearly and system specifically on an atomic scale (see Figure 3.9). More generally, one could say that the breakdown of the linear response approximation is connected with the fact that the liquid medium is structured [105],... 

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