Wavefunction parameters

The intra-orbit optimization process is carried out by varying the density parameters uj, 6,-, cj while keeping the wavefunction (or orbital) parameters fixed. Alternatively, inter-orbit optimization is accomplished by optimizing the wavefunction parameters for the energy functional... 

This expression depends on the electronic wavefunction parameters, A, and for a MCSCF electronic wavefunction, we have... 

The optimization of the wavefunction parameters is obtained by expanding the energy functional to second order in the nonredundant electronic parameters. The symbol Aw denotes the electronic wavefunction parameters at the Ath iteration. We write the energy difference to second order in the parameters between the Ath iteration and the next optimization step as... 

We wish to expand this expression in terms of the wavefunction parameters and collect terms that are linear with respect to S(t) and k( ). For this we define the following states... 

Define a physically reasonable and simple, but to a certain extent arbitrary, functional relation between the wavefunction parameters and the perturb-ational parameters. This dependence should be chosen so that the quality of the description of the system is approximately the same with or without the perturbation. 

A single solution of the response equation (16) should cost about as much as the determination of the wavefunction at a single geometry. Indeed, Eq. (16) is nothing else than the equation for the wavefunction parameters at an infinitesimally displaced geometry. The formal solution of Eq. (16) is... 

As is well-known, the time-dependent variational principle (TDVP) applied to the quantum mechanics action, when fully general variations in state vector space are possible, yields the time-dependent Schrodinger equation. However, when the variations take place in a limited space determined by the choice of an approximate form of wavefunction the result is a set of coupled first-order differentiS equations that govern the time-evolution of the wavefunction parameters (27). 

A quantum mechanical treatment of molecular systems usually starts with the Bom-Oppenheimer approximation, i.e., the separation of the electronic and nuclear degrees of freedom. This is a very good approximation for well separated electronic states. The expectation value of the total energy in this case is a function of the nuclear coordinates and the parameters in the electronic wavefunction, e.g., orbital coefficients. The wavefunction parameters are most often determined by the variation theorem the electronic energy is made stationary (in the most important ground-state case it is minimized) with respect to them. The... 

The spectral representations above are not computationally efficient, as they would require knowledge of all intermediate excited states. Computationally tractable formulas for the response functions within the various approximate methods are obtained instead through the following steps (1) choose a time-independent reference wavefunction (2) choose a parametrization of its time-development, for instance an exponential parametrization (3) set up the appropriate equations for the time development of the chosen wavefunction parameters (4) solve these equations in orders of the perturbation to obtain the wavefunction (parameters) (5) insert the solutions of these equations into the expectation value expression and obtain the RTFs and (6) identify the excited-state properties from the poles and residues. The computationally tractable formulas for the response functions therefore differ depending on the electronic structure method at hand, and the true spectral representations given above are only valid in the limit of a frill-configuration interaction (FCI) wavefunction. For approximate methods (i.e., where electron correlation is only partially included), matrix equations appear instead of the SOS expressions, for example. 

In addition to the wavefunction parameters, c p and Cofe, also the basis functions Xfi can depend on the perturbation. This will be the case when the perturbation... 

This is always the case for a SCF and MCSCF wavefunction, because they are optimized with respect to all wavefunction parameters. Truncated Cl wavefimctions, by contrast, are not variationally optimized with respect to the molecular orbital coefficients. The Hellmann-Feynman theorem is therefore satisfied only in the limit of a full Cl wavefunction, when the molecular orbital coefficients are redundant. 

In the case of the coupled cluster wavefunction the equations for the wavefunction parameters, i.e. for the coupled cluster amplitudes are simply the equations for the coupled cluster vector function in Eq. (9.81). The constraints are then = 0 and the coupled cluster Langrangian (Christiansen et ai, 1995a, 19986) is given as... 

Inserting the expression for the time-dependent MCSCF wave function, Eq. (11.36), and the perturbation expansion of the wavefunction parameters, Eq. (11.39), and separating the orders one finds for the first-order equation [see Exercise 11.5]... 

The Hellmann-Feynman theorem (J. Hellmann. Einfiihrung in die Quantenchemie , Deuticke, Leipzig, 1937 R. P, Feynman, Phys. Rev, 1939, 56, 340-343) states that first-oider properties can be evaluated as simple expectation values of the unperturbed wavefunction over the corresponding perturbed operator V. However, it should be noted that the Hellmann-Feynman theorem is not satisfied for truncated CC approaches and that evaluation of the energy derivatives appears to be the preferred choice. Furthermore, in ca.ses where additional wavefunction parameters such as, for example, the supplied basis function, exhibit a perturbation dependence, the Hellmann-Feynman theorem is not valid as long us finite basis sets are used. 

Table 1 The Orders of the Responses Needed to Calculate the Energy to a Given Order in the Perturbation According to the 2n + I Rule for the Wavefunction Parameters and the 2n+2 rule for the Lagrange Multipliers...

For variational wavefunctions, the wavefunction parameters obey the 2ri + I rule, which states that the derivatives of the parameters to order n determine the derivatives of the energy to order 2n + I. The same rule is obeyed also by the wavefunction parameters of nonvariational wavefunctions, whereas the Lagrange multipliers follow the 2n +2 rule. 

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