Electronic wave functions, stationary

Boys, S. F., Proc. Roy. Soc. London) A200, 542, Electronic wave function. I. A general method of calculation for the stationary state of any molecular systems." a. 

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. 

An important difference between the BO and non-BO internal Hamiltonians is that the former describes only the motion of electrons in the stationary field of nuclei positioned in fixed points in space (represented by point charges) while the latter describes the coupled motion of both nuclei and electrons. In the conventional molecular BO calculations, one typically uses atom-centered basis functions (in most calculations one-electron atomic orbitals) to expand the electronic wave function. The fermionic nature of the electrons dictates that such a function has to be antisymmetric with respect to the permutation of the labels of the electrons. In some high-precision BO calculations the wave function is expanded in terms of basis functions that explicitly depend on the interelectronic distances (so-called explicitly correlated functions). Such... 

The starting point of the creation of the theory of the many-electron atom was the idea of Niels Bohr [1] to consider each electron of an atom as orbiting in a stationary state in the field, created by the charge of the nucleus and the rest of the electrons of an atom. This idea is several years older than quantum mechanics itself. It allows one to construct an approximate wave function of the whole atom with the help of one-electron wave functions. They may be found by accounting for the approximate states of the passive electrons, in other words, the states of all electrons must be consistent. This is the essence of the self-consistent field approximation (Hartree-Fock method), widely used in the theory of many-body systems, particularly of many-electron atoms and ions. There are many methods of accounting more or less accurately for this consistency, usually named by correlation effects, and of obtaining more accurate theoretical data on atomic structure. 

The quantum theory of molecular structure developed here and the standard BO approach rely on the separability between electronic and nuclear configurational degrees of freedom. However, the way this is achieved differs radically between the approaches. In the treatment described here, the nuclei are seen to be trapped by an attractor generated by the stationary electronic wave function (nuclei follow the electronic states ) the electronic wave function does not depend upon the instantaneous positions of the nuclei as early proposed by this author [4] a change of electronic state, characterizing a chemical reaction with reactants and products in their ground electronic states, is described as a Franck-Condon like process. 

The separability used here leads to a clear relationship between chemical species and ground state electronic wave functions. Each isomeric species is determined by its own stationary ground state electronic wave function. The latter determines a stationary arrangement of Coulomb sources which is different for the different isomers. The nuclei are then hold around a stationary configuration if eq.(10) has bound solutions. An interconversion between them would require a Franck-Condon process, as it is discussed in Section 4. 

Generally speaking, the adiabatic wave function (2) is not a stationary one because it is not the eigen function of total Hamiltonian of the system (1). In reality, the electron wave function J/M(r R) depends on R and so the differential operator rR acts not only on / (R), but also on i/q/r R). It results in appearance of non-adiabatic correction operator in the basis of functions (2)... 

The field gradient is measured at a fixed point within the molecule, the translational part of the wave-function is thus of no consequence for (qap)-The effect of molecular rotation does, however, modify (qap) but the relationship between the rotating and stationary (qap) s has already been treated in the chapter dealing with microwave spectroscopy. In the present context, we are interested in the field gradients in a vibrating molecule in a fixed coordinate system. The Born-Oppenheimer approximation for molecular wave-functions enables us to separate the nuclear and electronic motions, the electronic wave functions being calculated for the nuclei in various fixed positions. The observed (qap) s will then be average values over the vibrational motion. 

If the electronic wave function (j>(Q, q) remains stationary during the time evolution of the entire polyatomic system, the expansion [Eq. (5)] of the total wave function reduces to a single term, i.e., the nuclear and electronic motion is... 

Equation (7) represents a crucial result for theoretical chemistry if the electronic wave function q) is stationary while the time evolution of the polyatomic system is in progress, the nuclei move in the field of force, the potential of which is equal to the energy of one of the eigenstates of the electronic subsystem. In this connection, the potential function Wm = Wm(Q) is referred to as the potential energy surface (PES) corresponding to the mth electronic state of the polyatomic system (10-12). 

In the next paragraph we will turn our attention to the examination of conditions implying stationary electronic wave functions 4>(Q, q). 

The Feynman diagram for the simplest annihilation event shows that annihilation is possible when the two particles are Ax h/mc 10 12 5 m apart, and that the duration of the event is At h/mc2 10-21 s. The distance is the geometric mean of nuclear and atomic dimensions, which is probably not significant. The distance is so much smaller than electronic wave functions that it may be assumed to be zero in computations of annihilation rates. The time is so short that, during it, a valence electron in a typical atom or molecule moves a distance of only ao/104, so that a spectator electron can be assumed to be stationary and the annihilating electron can be assumed to disappear in zero time. Thus the calculation of annihilation rates requires the evaluation of expectation values of the Dirac delta function, and the relaxation of the daughter system (post-annihilation remnant) can be understood with the aid of the sudden approximation [4], These are both relatively simple computations, providing an accurate wave function is available. 

Quantum mechanics (QM) is a field of quantum chemistry that uses mathematical basis to study chemical phenomena at a molecular level. It uses a complex mathematical expression called as a wave function with which energy and properties of atoms and molecules can be computed. For simple model systems wave functions can be analytically determined, while for complex systems such as those that involve molecular modeling, approximations have to be made. One of the commonly employed approximation methods is that of Born and Oppenheimer. This approximation exploits the idea that does not necessitate the development of a wave function description for both the electrons and the nuclei at the same time. The nuclei are heavier, and move much more slowly than the electrons, and therefore can be regarded as stationary, while electronic wave function is computed. By computing the QM of the electronic motion, the energy changes for different chemical processes, vibrations and chemical reactions can be understood.142... 

For a stationary, time-independent equilibrium structure (in the quantum chemical sense) all these effects can in principle be captured at once by the electronic wave function... 

The distribution of electric charge in a molecule is intimately related to its structure and reactivity. Knowledge of the distribution gives us a feeling for the physical and chemical properties of the molecule and provides a valuable assessment of the accuracy of approximate molecular wavefunctions. The charge distribution in the nth stationary state is determined by the many-electron wave function 0 of the free molecule. If the molecule interacts with an external electric perturbation E, the wave function determines the distortion and,... 

Two isomers can have now a well-defined stationary arrangement of external Coulomb sources, say aoi and aok, respectively. Each one has a different Schrodinger equation (8) wherefrom the electronic wave function can be determined with modem analytical gradient techniques. The problem now is to find solutions for the molecular problem, eq(4). 

The electronic wave function for the i-th stationary state is only tied to the external sources of the potential aoi, namely, it is independent of the point chosen in the nuclear configuration space. Using the completeness of the solutions to eq.(8) one can write ... 

This is an eigenvalue equation for the nuclei degrees of freedom submitted to the model potential generated by the exact electronic wave function Yj(p aoi). Depending upon the particular mass vector, the electronic potential may sustain bound nuclear dynamical states of diverse frequencies. The nuclear stationary states Xik(R) in the inertial frame are obtained as functions relative to the electronic attractor potential Ej(R). 

In the R-BO scheme, the stationary electronic wave function drives the nuclear dynamics via the setup of a fundamental attractor acting on the sources of Coulomb field [11]. The nuclei do not have an equilibrium configuration as they are described as quantum systems and not as classical particles. The concept of molecular form (shape) is related to the existence of stationary nuclear state setup by the electronic attractor and their interactions with external electromagnetic fields. 

A rigged BO approach is developed and used to describe a chemical system calculated with present day advanced electronic methods. Chemical species are determined by electronic wave functions that are independent from the nuclear configuration space. This is the fundamental hypothesis [11]. Boundary conditions in the global electronic wave function are introduced via the solution of electronic Schrodinger equations for systems of external Coulomb sources (Cf. Eq.(8)). The associated stationary arrangement of external Coulomb sources allows for the introduction of molecular frames. This approach naturally leads to a state-to-state description particularly useful in gas phase reactions. A chemical reaction is described as if it were an electronic spectroscopy event or series of events. 

In summary, the rigged Born-Oppenheimer framework permits a general description of chemical reactions. By retaining the stationary geometry structures determined with modern electronic wave function methods the relationship between quantum electronic state and molecular species is established. The relaxation processes involve serial changes of quantum states. 

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