Wave functions and orbitals

Section 1 1 A review of some fundamental knowledge about atoms and electrons leads to a discussion of wave functions, orbitals, and the electron con figurations of atoms Neutral atoms have as many electrons as the num ber of protons m the nucleus These electrons occupy orbitals m order of increasing energy with no more than two electrons m any one orbital The most frequently encountered atomic orbitals m this text are s orbitals (spherically symmetrical) and p orbitals ( dumbbell shaped)... 

Photoexcitation may be useful to drive chemical reactions, for example, photoin-duced electron transfer and cis-trans isomerization. In the great majority of cases, aromatic i-systems are excited or deexcited, but the reaction may involve a rotation around a single bond. Not surprisingly, the products are often different from the products of a thermal reaction. These differences can be explained by the wave functions (orbitals) and states. 

As is well known, when the electronic spin-orbit interaction is small, the total electronic wave function v / (r, s R) can be written as the product of a spatial wave function R) and a spin function t / (s). For this, we can use either... 

HyperClicm cati plot orbital wave fuuctious resulting fmni serni-cmpirical and ah i/iitw quan tii m m ecli an ica I calculations. It is ill tercstiu g to view both tli c u tidal properties an d th e relative sizes of the wave functions. Orbital wave functiou s can provide dietni-cal in sigh is. 

HyperChem can plot orbital wave functions resulting from semi-empirical and ab initio quantum mechanical calculations. It is interesting to view both the nodal properties and the relative sizes of the wave functions. Orbital wave functions can provide chemical insights. 

In the three following sections we will try to sketch the mathematical foundation for the three approaches which are most closely connected with the Hartree-Fock scheme, namely the methods of superposition of configurations (a), correlated wave functions (b), and different orbitals for different spins (c). We will also discuss their main physical implications. 

The shapes of atomic orbitals are routinely confused with graphs of the angular factors in wave functions [60] and shown incorrectly. The graph of a py orbital, for example, gives tangent spheres lying on the y-axis. 

The quantum numbers n and i. Multi-electron atoms can be characterized by a set of principal and orbital quantum numbers n, t which labels one-electron wave functions (orbitals). 

Clearly, however, electrons exist. And they must exist somewhere. To describe where that somewhere is, scientists used an idea from a branch of mathematics called statistics. Although you cannot talk about electrons in terms of certainties, you can talk about them in terms of probabilities. Schrodinger used a type of equation called a wave equation to define the probability of finding an atom s electrons at a particular point within the atom. There are many solutions to this wave equation, and each solution represents a particular wave function. Each wave function gives information about an electron s energy and location witbin an atom. Chemists call these wave functions orbitals. 

In further studies of chemistry and physics, you will learn that the wave functions that are solutions to the Schrodinger equation have no direct, physical meaning. They are mathematical ideas. However, the square of a wave function does have a physical meaning. It is a quantity that describes the probability that an electron is at a particular point within the atom at a particular time. The square of each wave function (orbital) can be used to plot three-dimensional probability distribution graphs for that orbital. These plots help chemists visualize the space in which electrons are most likely to be found around atoms. These plots are... 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

Size consistency The DMRG ansatz is size-consistent when using a localized basis (e.g., orthogonalized atomic orbitals) in which the wave function for the separated atoms can be considered to factorize into the wave functions for the individual atoms expressed in disjoint subsets of the localized basis. To see this in an informal way, let us assume that we have two DMRG wave functions Pa) and I Pg) for subsystems A and B separately. Both Pa and VPB have a matrix product structure, that is... 

For a multi-Slater determinant wave function, orbitals which satisfy Eq. (3.6), and therefore Eq. (3.7), can still be defined. For these orbitals, referred to as the natural spin orbitals, the coefficients nt are not necessarily integers, but have the boundaries 0 n, 1. 

The minimum requirements for a many-electron wave function, namely, antisymmetry with respect to interchange of electrons and indistinguishability of electrons, are satisfied by an antisymmetrized sum of products of one-electron wave functions (orbitals), ( 1),... 

In summary, to obtain a many-electron wave function of the single determinantal form [equation (A.12)] which will give the lowest electronic energy [equation (A.14) or (A.27)], one must use one-electron wave functions (orbitals) which are eigenfunctions of the one-electron Fock operator according to equation (A.42). There are many, possibly an infinite number of, solutions to equation (A.42). We need the lowest Ne of them, one for each electron, for equation (A. 12) [or (A.27)]. When the Ne MOs of lowest energy satisfy equation (A.42), then Eq=Ehf [equation (A.27)] and o= hf [equation (A.12)]. 

Thus, let us assume that two different external potentials can each be consistent with the same nondegenerate ground-state density po- We will call these two potentials Va and wj, and the different Hamiltonian operators in which they appear and Ht,. With each Hamiltonian will be associated a ground-state wave function Pq and its associated eigenvalue Eq. The variational theorem of molecular orbital theory dictates that the expectation value of the Hamiltonian a over the wave function b must be higher than the ground-state energy of a, i.e.. 

Derivation of Results about Tetrahedral Orbitals.—The results about tetrahedral bond orbitals described. above are derived in the following way. We assume that the radial parts of the wave functions fa and faxf faM are so closely similar that their differences can be neglected. The angular parts are... 

Because no symmetry operation can alter the value of R(n, r), we need not consider the radial wave functions any further. Symmetry operations do alter the angular wave functions, however, and so we shall now examine them in more detail. It should be noted that, since A(0, 0) does not depend on n, the angular wave functions for all s, all / , all d, and so on, orbitals of a given type are the same regardless of the principal quantum number of the shell to which they belong. Table 8.1 lists the angular wave functions for sy p, d, and / orbitals. 

Suppose we have two isolated hydrogen atoms. We may describe tbetn by the wave functions ( /A and 0B, each having the form given in Chapter 2 for a is orbital. If the atoms are sufficiently isolated so that they do not interact, the wave function for the system of two atoms is... 

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