Wave function general

Exact or multi-determinant wave function (general, electronic, nuclear)... 

The summed weights, classified as covalent and ionic, show a consistent trend with the nature of the X atom. The value of the covalent contribution to the total VB wave function generally increases as X gets heavier from Si —> Pb for the CH3—XH3 series, and from... 

Shore et al. [4] and Wagner et al. [5] proved for exciton or dimer models coupled to one phonon mode that the two-center wave function generalized to an asymmetric nonunitary ansatz with a VP 17 of the form... 

With regard to the second point, it is important to note that an approximate wave function which is more general than those of Eqs. (2)-(5) cannot be described in terms of either bent bonds or wave function (general MCSCF, GVB-CI or Cl) will be a complicated combination of these two descriptions (as well as others, e.g., the atoms-in-molecule picture (10)) or in certain approximate wave functions the descriptions are related by a transformation and are thus in some sense equivalent (10). Hence the best one can do is decide on a criterion to measure the extent to which a particular picture is contained in the general wave function. One possible measure would be the overlap of a unique or unique bent bond description with the general wave function. 

Hartke, B. and E. A. Carter (1992). Ab Initio Molecular Dynamics with Correlated Molecular Wave Functions Generalized Valence Bond Molecular Dynamics and Simulated Annealilng. J. Chem. Phvs. 97(9) 6569-6578. 

In eqn (4.68), we developed the tight-binding formalism for periodic systems in which there was only one atom per unit cell, with each such atom donating n basis functions to the total electronic wave function. Generalize the discussion presented there to allow for the presence of more than one atom per cell. In concrete terms, this means that the basis functions acquire a new index, i, J, a), where the index i specifies the unit cell, / is a label for the atoms within a unit cell and a remains the label of the various orbitals on each site. 

Employing the superposition principles with periodic constrains to formulate the wave function general forms specific to solid state -the Bloch theorem and orbitals ... 

In this seiniclassical calculation, we use only one wavepacket (the classical path limit), that is, a Gaussian wavepacket, rather than the general expansion of the total wave function. Equation (39) then takes the following form ... 

Section IB presents results that the analytic properties of the wave function as a function of time t imply and summarizes previous publications of the authors and of their collaborators [29-38]. While the earlier quote from Wigner has prepared us to expect some general insight from the analytic behavior of the wave function, the equations in this secbon yield the specific result that, due to the analytic properties of the logarithm of wave function amplitudes, certain forms of phase changes lead immediately to the logical necessity of enlarging... 

In the same section, we also see that the source of the appropriate analytic behavior of the wave function is outside its defining equation (the Schibdinger equation), and is in general the consequence of either some very basic consideration or of the way that experiments are conducted. The analytic behavior in question can be in the frequency or in the time domain and leads in either case to a Kramers-Kronig type of reciprocal relations. We propose that behind these relations there may be an equation of restriction, but while in the former case (where the variable is the frequency) the equation of resh iction expresses causality (no effect before cause), for the latter case (when the variable is the time), the restriction is in several instances the basic requirement of lower boundedness of energies in (no-relativistic) spectra [39,40]. In a previous work, it has been shown that analyticity plays further roles in these reciprocal relations, in that it ensures that time causality is not violated in the conjugate relations and that (ordinary) gauge invariance is observed [40]. 

When the wave function is completely general and pennitted to vary in the entire Hilbert space the TDVP yields the time-dependent Schrodinger equation. However, when the possible wave function variations are in some way constrained, such as is the case for a wave function restricted to a particular functional form and represented in a finite basis, then the corresponding action generates a set of equations that approximate the time-dependent Schrodinger equation. 

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... 

Better END approximations are defined by the introduction of more general molecular wave functions leading to larger and more involved parameter spaces. 

When constructing more general molecular wave functions there are several concepts that need to be defined. The concept of geometry is inhoduced to mean a (time-dependent) point in the generalized phase space for the total number of centers used to describe the END wave function. The notations R and P are used for the position and conjugate momenta vectors, such that... 

The product analysis of the END system wave function is quite general, but for simplicity wc consider the case of Iw o product rragiriciits. A and B. As these... 

In this chapter, we look at the techniques known as direct, or on-the-fly, molecular dynamics and their application to non-adiabatic processes in photochemistry. In contrast to standard techniques that require a predefined potential energy surface (PES) over which the nuclei move, the PES is provided here by explicit evaluation of the electronic wave function for the states of interest. This makes the method very general and powerful, particularly for the study of polyatomic systems where the calculation of a multidimensional potential function is an impossible task. For a recent review of standard non-adiabatic dynamics methods using analytical PES functions see [1]. 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

Inserting this completely general wave function into Eq. (B.l), multiplying by exp(— S(f )), and separating the real and imaginary parts leads to... 

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