Wave functions variational parameter

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). 

For a variationally optimized wave function, the first term is again zero (eq. (10.24)). Furthermore, the second term, which involves calculation of the second derivative of the wave function with respect to the parameters, can be avoided. This can be seen by differentiating the stationary condition (10.24) with respect to the perturbation. 

The idea is to construct a Lagrange function which has the same energy as the non-variational wave function, but which is variational in all parameters. Consider for example a CL wave function, which is variational in the state coefficients (a) but not in the MO coefficients (c) (note that we employ lower case c for the MO coefficients, but capital C to denote all wave function parameters, i.e. C contains both a and c), since they are determined by the stationary condition for the HF wave function. 

The energy of a wave function containing variational parameters, like an HF (one Slater determinant) or MCSCF (many Slater determinants) wave function. Parameters are typically the MO and state coefficients, but may also be for example basis function exponents. Usually only minima are desired, although in some cases saddle points may also be of interest (excited states). 

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

As required by (36), the variational parameter k is calculated to vary between k = 2 at R = 0 and k = 1 at R > 5ao- The parameter k is routinely interpreted as either a screening constant or an effective nuclear charge, as if it had real physical meaning. In fact, it is no more than a mathematical artefact, deliberately introduced to remedy the inadequacy of hydrogenic wave functions as descriptors of electrons in molecular environments. No such parameter occurs within the Burrau [84] scheme. 

Although the calculated molecular parameters De = 3.15 eV, re = 1.64 a0 do not compare well with experiment the simplicity of the method is the more important consideration. Various workers have, for instance, succeeded to improve on the HL result by modifying the simple Is hydrogenic functions in various ways, and to approach the best results obtained by variational methods of the James and Coolidge type. It can therefore be concluded that the method has the correct symmetry to reproduce the experimental results if atomic wave functions of the correct form and symmetry are used. The most important consideration will be the effect of the environment on free-atom wave functions. 

The virial ratio is, as we noted above, 1.3366 for the separate-atom AO basis MO calculation, i.e. not 1.0. Now within the confines of the linear variation method (the usual LCAO approach) there is no remaining degree of freedom to use in order to constrain the virial ratio to its formally correct value (or indeed to impose any other constraint). Thus imposing the correct virial ratio on the linear variation method is, in this case, not possible without simultaneously destroying the symmetry of the wave function. Only by optimising the non-linear parameters can we improve the virial ratio as the above results show. Even at this most elementary level, the imposition of various formally correct constraints on the wave function is seem to generate contradictions. 

The conclusion above that optimisation of the non-linear parameters in the AO basis leads to a basis with correct spatial symmetry properties cannot be true for all intemuclear separations. At R = 0 the orbital basis must pass over into the double-zeta basis for helium i.e. two different 1 s orbital exponents. It would be astonishing if this transition were discontinuous at R = 0. While considering the variation of basis with intemuclear distance it is worth remembering that the closed-shell spin-eigenfunction MO method does not describe the molecule at all well for large values of R the spin-eigenfunction constraint of two electrons per spatial orbital is completely unrealistic at large intemuclear separation. With these facts in mind we have therefore computed the optimum orbital exponents as a function of R for three wave functions ... 

Each wave function was optimized with respect to the parameters L, s, and Ck. This lead to -m(m + 1) + 3 + 1 variational parameters per basis function (23 for... 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

The inversion operator i acts on the electronic coordinates (fr = —r). It is employed to generate gerade and ungerade states. The pre-exponential factor, y is the Cartesian component of the i-th electron position vector (mf. — 1 or 2). Its presence enables obtaining U symmetry of the wave function. The nonlinear parameters, collected in positive definite symmetric 2X2 matrices and 2-element vectors s, were determined variationally. The unperturbed wave function was optimized with respect to the second eigenvalue of the Hamiltonian using Powell s conjugate directions method [26]. The parameters of were... 

As we have written it in Equation (8), the DMRG wave function contains redundant variational parameters. This means that the set of variational tensors f/ni... ij/Hk in the DMRG wave function is not unique, because we can find another set of tensors whose matrix product yields an identical state. This redundancy is analogous to the redundancy of the orbital parametrization of the Hartree-Fock determinant. In the case of the DMRG wave function, we can insert a matrix T and its inverse between any two variational tensors and leave the state invariant... 

The parameter X has been embedded in the definition of Hp. The wave function from perturbation theory [equation (A.109)] is not normalized and must be renormalized. The energy of a truncated perturbation expansion [equation (A.110)] is not variational, and it may be possible to calculate energies lower than experimental. ... 

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