Wave functions of a molecule

As pointed out in the previous paragraph, the total wave function of a molecule consists of an electronic and a nuclear parts. The electrons have a different intrinsic nature from nuclei, and hence can be treated separately when one considers the issue of permutational symmetry. First, let us consider the case of electrons. These are fermions with spin and hence the subsystem of electrons obeys the Fermi-Dirac statistics the total electronic wave function... 

Consider either the electronic wave functions or the vibrational wave functions of a molecule. These functions satisfy the Schrodinger equation... 

The complete wave function of a molecule is called the rovibronic wave function. In the simplest approximation, the rovibronic function is a product of rotational, vibrational, and electronic functions. For certain applications, the rotational motion is first neglected, and the vibrational and electronic motions are treated together. The rotational motion is then taken into account. The wave function for electronic and vibrational motion is called the vibronic wave function. Just as we separately classified the electronic and vibrational wave functions according to their symmetries, we can do the same for the vibronic functions. In the simplest approximation, the vibronic wave function is a product of electronic and vibrational wave functions, and we can thus readily determine its symmetry. For example, if the electronic state is an e2 state and the vibrational state is a state, then the vibronic wave function is... 

This phenomenon of vibronic coupling can be treated very effectively by using group theoretical methods. As will be shown in Chapter 10, the vibrational wave function of a molecule can be written as the product of wave functions for individual modes of vibration called normal modes, of which there will be 3n - 6 for a nonlinear, /i-atomic molecule. That is, we can... 

The complete wave function of a molecule will often change in the presence of foreign fields. Foreign fields, either electrostatic or electromagnetic, may under certain conditions change wave functions sufficiently to permit perturbations which would otherwise be unimportant. Perturbations may also be induced by collisions with other molecules, particularly with molecules which are themselves paramagnetic. These effects give rise to what is often called collision-induced predissociation. 

The coupled-cluster (CC) method [41, 118-120] expresses the wave function of a molecule in the ground state by... 

Davies[190] The problem is to explain why it is that in most situations. . . the wave function of a molecule seems to be not an eigenstate of the Hamiltonian, but one of a class of slowly time varying states which are more stable in some sense. 

The complete A -electron wave function of a molecule is represented in the form of the Slater determinant composed of the MOs ... 

Thus we came to the conclusion that in the MO method the total (7V-electron) wave function of a molecule is expressed via one-electron functions, that is, molecular spin orbitals which, in their turn, are expressed via atomic spin orbitals. Subsequently we shall be utilizing only MOs expressed in the LCAO form. Although the spin variables will not be taken into account, the Pauli principle will be obeyed. 

The approximation involved in factorization of the total wave function of a molecule into electronic, vibrational and rotational parts is known as the Bom-Oppenheimer approximation. Furthermore, the Schrodinger equation for the vibrational wave function (which is the only part considered here), transformed to the normal coordinates Qi (which are linear functions of the "infinitesimal displacements q yields equations of the harmonic oscillator t5q>e. For these reasons Lifson and Warshel have stressed that the force-field calculations should not be considered as classical-me-... 

The electron denstiy distribution p(r) of an atom or molecule is an observable property that can be measured by a combination of X-ray and neutron diffraction experiments [22]. Also, it is easy to calculate p(r) once the MOs and the wave function of a molecule have been determined. The distribution p(r) is invariant with regard to any unitary transformation of the MOs. It has been shown by Hohenberg and Kohn that the energy of a molecule in its (nondegenerate) ground state is a unique functional of p(r) [23]. In other words, the physical and chemical properties of a molecule can be related to p(r). Thus, p(r) represents the best starting point for an analysis of chemical bonding. 

Symmetry operators can operate on functions as well as on points and can have eigenfunctions with eigenvalues equal to 1 or to —1. An electronic wave function of a molecule can be an eigenfunction of the synunetry operators which belong to the nuclear framework of a molecule. 

Pauling. In the method of molecular orbitals, the wave function of a molecule may be set up as a product of molecular orbitals, a molecular orbital being the wave function of a single electron in the... 

In the Bom-Oppenheimer approximation the total vibronic wave function of a molecule may be written as a product function ... 

The electronic wave function of a molecule containing N electrons depends on 3N Cartesian coordinates of the electrons and on their N spin coordinates (for each electron, its o = or — ). Thus, it is a function of position in 4A-dimensional space. This function will be created out of simple bricks , i.e., molecular spinorbitals. Each of those will be a function of the coordinates of one electron only three Cartesian coordinates and one spin coordinate (cf.. Chapter 1). A spinorbital is therefore a function of the coordinates in the 4-D space, and in the most general case, a normalized spinorbital reads as (see Fig. 8.1) ... 

The total wave function of a molecule comprising N electrons and M nuclei depends on all electronic rj and nuclear Rj coordinates. It can be obtained by solving the time-independent Schrodinger equation [14-16],... 

This chapter discusses theorems that are used in molecular quantum mechanics. Section 14.1 expresses the electron probability density in terms of the wave function. Section 14.2 shows how the dipole moment of a molecule is calculated from the wave function. Section 14.3 gives the procedure for calculating the Hartree-Fock wave function of a molecule. Sections 14.4 to 14.7 discuss the virial theorem and the Hellmann-Feynman theorem, which are helpful in understanding chemical bonding. 

In H2, the two nuclear spins of the H atoms can be combined in two different ways, resulting in the molecular species with the total nuclear spin of the two hydrogens of / = 1 or / = 0. In the ground electronic and vibrational state, each H2 molecule can be characterized by a combination of a certain rotational state with a certain nuclear spin state. As protons are fermions, Pauli s principle requires that the total wave function of a molecule should be antisymmetric with respect to the permutation of the two nuclei. The rotational states with even values of the rotational quantum number / are symmetric with respect to such permutations, including the rotational state with the lowest energy, which has / = 0. Such rotational states can be combined only with the antisymmetric nuclear spin state with 1 = 0, and such combinations correspond to parahydrogen (PH2). All rotational states with odd / values are antisymmetric and are only allowed in combination with the symmetric (/ = 1) nuclear spin states. These combinations correspond to orthohydrogen (oH2). At the same time, H2 molecules with even-even and odd-odd combinations of the two quantum numbers do not exist. As a result. 

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