Cartesian coordinates electronic states

In principal one can calculate the electronic energy as a function of the Cartesian coordinates of the three atomic nuclei of the ground state of this system using the methods of quantum mechanics (see Chapter 2). (In subsequent discussion, the terms coordinates of nuclei and coordinates of atoms will be used interchangeably.) By analogy with the discussion in Chapter 2, this function, within the Born-Oppenheimer approximation, is not only the potential energy surface on which the reactant and product molecules rotate and vibrate, but is also the potential... 

The function F(l,2) is in fact the space part of the total wave function, since a non-relativistic two-electron wave function can always be represented by a product of the spin and space parts, both having opposite symmetries with respect to the electrons permutations. Thus, one may skip the spin function and use only the space part of the wave function. The only trace that spin leaves is the definite per-mutational symmetry and sign in Eq.(14) refers to singlet as "+" and to triplet as Xi and yi denote cartesian coordinates of the ith electron. A is commonly known angular projection quantum number and A is equal to 0, 1, and 2 for L, II and A symmetry of the electronic state respectively. The linear variational coefficients c, are found by solving the secular equations. The basis functions i(l,2) which possess 2 symmetry are expressed in elliptic coordinates as ... 

The simple harmonic motion of a diatomic molecule was treated in Chapter 1, and will not be repeated here. Instead, emphasis is placed on polyatomic molecules whose electronic energy s dependence on the 3N Cartesian coordinates of its N atoms can be written (approximately) in terms of a Taylor series expansion about a stable local minimum. We therefore assume that the molecule of interest exists in an electronic state for which the geometry being considered is stable (i.e., not subject to spontaneous geometrical distortion). 

The explicit dependence on R is not shown for equation (A.6) and will not be given in subsequent equations, it being understood that unless stated otherwise, we are working within the BO approximation. The Laplacian operator V(/)2 in Cartesian coordinates for the 7th electron is given by... 

The parity of atomic states is important in spectroscopy. A radial function is an even function [see (1.113)] the spherical harmonic Y(m is found to be an even or odd function of the Cartesian coordinates according to whether / is an even or odd number. For a many-electron atom, it follows that states arising from a configuration for which the sum of the / values of all the electrons is an even number are even functions when 2,/, is odd, the state has odd parity. 

Specify geometry, charge and electronic state, e.g. CH4 cartesian coordinates, charge = 0, singlet or CH4 cartesian coordinates, charge = 0, triplet, etc. 

The variables describing electrons and nuclei are termed electronic and nuclear. For the majority of problems which arise in chemistry, the nuclear variables can be thought to be the Cartesian coordinates of the nuclei in the physical three-dimensional space. Of course the nuclei are in fact inherently quantum objects which manifest in such characteristics as nuclear spins - additional variables describing internal states of nuclei, which do not have any classical analog. However these latter variables enter into play relatively rarely. For example, when the NMR, ESR or Mossbauer experiments are discussed or in exotic problems like that of the ortho-para dihydrogen conversion. In a more common setting, such as the one represented by the... 

An LCAO description of the electronic structure requires at least the minimal basis set (all orbitals that may be occupied in the ground state of the atom) of five d states per atom and the s state. Consideration of the bands from F ig. 20-1 indicates that in fact the highest-cnergy slates shown (for examples, H,s and X4) have p-like symmetry, and we shall not reproduce this with our minimal set, but the bands at this energy arc unoccupied in any case and it will be of little consequence. For constructing bands in solids, the angular forms for the d states in terms of cartesian coordinates, shown in Eq. (1-21), arc most convenient. Here we shall carry out the calculation explicitly for chromium, in the body-centered cubic structure it is carried out for the face-centered cubic structure in Problem... 

We wish to show first that the electron clouds of the three p electrons, possessing a shape similar to a figure eight, are directed along the cartesian coordinates. If / (r) represents the radial part of the wave function, we obtain from equation 18.14 and 18.34 the following expressions for the three p states ... 

The simplest way to combine electronic stnicture calculations with nuclear dynamics is to use harmonic analysis to estimate both vibrational averaging effects on physico-chemical observables and reaction rates in terms of conventional transition state theory, possibly extended to incorporate tunneling corrections. This requires, at least, the knowledge of the structures, energetics, and harmonic force fields of the relevant stationary points (i.e. energy minima and first order saddle points connecting pairs of minima). Small anq)litude vibrations around stationary points are expressed in terms of normal modes Q, which are linearly related to cartesian coordinates x... 

The energy of the conduction electrons is given by h2k2/2 x, where k is the wave vector number and p, is the effective mass of the electron-nucleus. In a real space of Cartesian coordinates k = [kx, ky, fcj, a Fermi sphere can be constructed with radius k = (2 lE )x 2/h. The shape of this sphere is a clearly defined by the electrical properties of the metal. The current density obeys the change in the occupancy of states near the Fermi level, which separates the unfilled orbitals in the metal from the filled ones in the linear momentum space p = hk. 

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