Wave function properties

Temperature, wave function properties, 214 Tetraatomic molecules ... 

Several desirable wave function properties are obtained with the use of STOs... 

However, although we will not directly solve this equation, we will determine the solutions by the test functions method. Thus, based on the stationary wave functions properties, to be continuous, derivable and tend to zero when the variable tends to infinite, one will try the wave function with the right form, which corresponds to the first vibration mode ... 

Small metal clusters are also of interest because of their importance in catalysis. Despite the fact that small clusters should consist of mostly surface atoms, measurement of the photon ionization threshold for Hg clusters suggest that a transition from van der Waals to metallic properties occurs in the range of 20-70 atoms per cluster [88] and near-bulk magnetic properties are expected for Ni, Pd, and Pt clusters of only 13 atoms [89] Theoretical calculations on Sin and other semiconductors predict that the stmcture reflects the bulk lattice for 1000 atoms but the bulk electronic wave functions are not obtained [90]. Bartell and co-workers [91] study beams of molecular clusters with electron dirfraction and molecular dynamics simulations and find new phases not observed in the bulk. Bulk models appear to be valid for their clusters of several thousand atoms (see Section IX-3). 

What is addressed by these sources is the ontology of quantal description. Wave functions (and other related quantities, like Green functions or density matrices), far from being mere compendia or short-hand listings of observational data, obtained in the domain of real numbers, possess an actuality of tbeir own. From a knowledge of the wave functions for real values of the variables and by relying on their analytical behavior for complex values, new properties come to the open, in a way that one can perhaps view, echoing the quotations above, as miraculous. ... 

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... 

The quantum phase factor is the exponential of an imaginary quantity (i times the phase), which multiplies into a wave function. Historically, a natural extension of this was proposed in the fonn of a gauge transformation, which both multiplies into and admixes different components of a multicomponent wave function [103]. The resulting gauge theories have become an essential tool of quantum field theories and provide (as already noted in the discussion of the YM field) the modem rationale of basic forces between elementary particles [67-70]. It has already been noted that gauge theories have also made notable impact on molecular properties, especially under conditions that the electronic... 

In addition, it can occasionally be useful to regard some physical parameter appearing in the theoi y as a complex quantity and the wave function to possess analytic properties with regard to them. This formal procedure might even include fundamental constants like e, h, and so on. 

They unfold a connection between parts of time-dependent wave functions that arises from the structure of the defining equation (2) and some simple properties of the Hamiltonian. 

The connection holds separately for the coefficient of each state component in the wave function and is not a property of the total wave function (as is, e.g., the dynamical phase [9]). 

U(qJ is referred to as an adiabatic-to-diabatic transformation (ADT) matrix. Its mathematical sbucture is discussed in detail in Section in.C. If the electronic wave functions in the adiabatic and diabatic representations are chosen to be real, as is normally the case, U(q ) is orthogonal and therefore has n n — l)/2 independent elements (or degrees of freedom). This transformation mabix U(qO can be chosen so as to yield a diabatic electronic basis set with desired properties, which can then be used to derive the diabatic nuclear motion Schrodinger equation. By using Eqs. (27) and (28) and the orthonormality of the diabatic and adiabatic electronic basis sets, we can relate the adiabatic and diabatic nuclear wave functions through the same n-dimensional unitary transformation matrix U(qx) according to... 

Although the leading term of the electronic wave function of the system is thus changed, the total wave function has not and the calculated trajectory and properties exhibit no discontinuous behavior. 

The concept of two-state systems occupies a central role in quantum mechanics [16,26]. As discussed extensively by Feynmann et al. [16], benzene and ammonia are examples of simple two-state systems Their properties are best described by assuming that the wave function that represents them is a combination of two base states. In the cases of ammonia and benzene, the two base states are equivalent. The two base states necessarily give rise to two independent states, which we named twin states [27,28]. One of them is the ground state, the other an excited states. The twin states are the ones observed experimentally. 

Symmetry considerations have long been known to be of fundamental importance for an understanding of molecular spectra, and generally molecular dynamics [28-30]. Since electrons and nuclei have distinct statistical properties, the total molecular wave function must satisfy appropriate symmehy... 

Let us examine a special but more practical case where the total molecular Hamiltonian, H, can be separated to an electronic part, W,.(r,s Ro), as is the case in the usual BO approximation. Consequendy, the total molecular wave function fl(R, i,r,s) is given by the product of a nuclear wave function X uc(R, i) and an electronic wave function v / (r, s Ro). We may then talk separately about the permutational properties of the subsystem consisting of electrons, and the subsystemfs) formed of identical nuclei. Thus, the following commutative laws Pe,Hg =0 and =0 must be satisfied X =... 

Let us discuss further the pemrutational symmetry properties of the nuclei subsystem. Since the elechonic spatial wave function t / (r,s Ro) depends parameti ically on the nuclear coordinates, and the electronic spacial and spin coordinates are defined in the BF, it follows that one must take into account the effects of the nuclei under the permutations of the identical nuclei. Of course. 

The Symmetry Properties of Wave Functions of Li3 Electronically Ground State in [Pg.582]

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