Solution to the Schrodinger equation

Soon we will need some basis functions that depend on the angles 0 and f , preferentially each of them somehow adapted to the problem we are solving. These basis functions will be generated as the eigenfunctions of Ti obtained at a fixed value 

The total wave function that also takes into account rotational degrees of freedom 0,4 ) is constructed as (the quantum number J = 0,1,2. determines the length of the angular momentum of the system, while the quantum number M = -J, gives the z component of the angular momentum) 

In what is known as the close coupling method the function from eq. (14.16) is inserted into the Schrodinger equation. Then, the resulting 

The summation extends over some assumed set of k, D, (the number of k, D, pairs is equal to the number of equations). The symbol o = (a, )8, y, 0, f ) means integration over the angles. The system of equations is solved numerically. 

when solving the equations, we apply the boundary conditions suitable for a discrete spectrum (vanishing for p = oo), we obtain the stationary states of the three-atomic molecule. We are interested in chemical reactions, in which one of the atoms comes to a diatomic molecule, and after a while another atom flies out leaving (after reaction) the remaining diatomic molecule. Therefore, we have to apply suitable boundary conditions. As a matter of fact we are not interested in details of the collision, we are positively interested in what comes to our detector from the spot where the reaction takes place. What may happen at a certain energy E to a given reactant state (i.e. what the product state is such a reaction is called state-to-state ) is determined by the corresponding o-(E). The cross 

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T indicates that the integration is over all space. Wavefunctions which satisfy this condition re said to be normalised. It is usual to require the solutions to the Schrodinger equation to be rthogonal ... 

In addition to initial conditions, solutions to the Schrodinger equation must obey eertain other eonstraints in form. They must be eontinuous funetions of all of their spatial eoordinates and must be single valued these properties allow T T to be interpreted as a probability density (i.e., the probability of finding a partiele at some position ean not be multivalued nor ean it be jerky or diseontinuous). The derivative of the wavefunetion must also be eontinuous exeept at points where the potential funetion undergoes an infinite jump (e.g., at the wall of an infinitely high and steep potential barrier). This eondition relates to the faet that the momentum must be eontinuous exeept at infinitely steep potential barriers where the momentum undergoes a sudden reversal. 

Approximation Methods Can be Used When Exact Solutions to the Schrodinger Equation Can Not be Eound. 

For small molecules, the accuracy of solutions to the Schrodinger equation competes with the accuracy of experimental results. However, these accurate ab initio calculations require enormous computation and are only suitable for the molecular systems with small or medium size. Ab initio calculations for very large molecules are beyond the realm of current computers, so HyperChem also supports semi-empirical quantum mechanics methods. Semi-empirical approximate solutions are appropriate and allow extensive chemical exploration. The inaccuracy of the approximations made in semi-empirical methods is offset to a degree by recourse to experimental data in defining the parameters of the method. Indeed, semi-empirical methods can sometimes be more accurate than some poorer ab initio methods, which require much longer computation times. 

Ab initio calculations can be performed at the Hartree-Fock level of approximation, equivalent to a self-consistent-field (SCF) calculation, or at a post Hartree-Fock level which includes the effects of correlation — defined to be everything that the Hartree-Fock level of approximation leaves out of a non-relativistic solution to the Schrodinger equation (within the clamped-nuclei Born-Oppenhe-imer approximation). 

For any but the smallest systems, however, exact solutions to the Schrodinger equation are not computationally practical. Electronic structure methods are characterized by their various mathematical approximations to its solution. There are two major classes of electronic structure methods ... 

Ab initio methods compute solutions to the Schrodinger equation using a series of rigorous mathematical approximations. These procedures are discussed in detail in Appendix A, The Theoretical Background. 

A theoretical model should be uniquely defined for any given configuration of nuclei and electrons. This means that specifying a molecular structure is all that is required to produce an approximate solution to the Schrodinger equation no other parameters are needed to specify the problem or its solution. 

Reproducing the exact solution for the relevant n-electron problem a method ought to yield the same results as the exact solution to the Schrodinger equation to the greatest extent possible. What this means specifically depends on the theory underlying the method. Thus, Hartree-Fock theory should be (and is) able to reproduce the exact solution to the one electron problem, meaning it should be able to treat cases like HeH ... 

Suppose we get a little more sophisticated about our question. The more advanced student might respond that the periodic table can be explained in terms of the relationship between the quantum numbers which themselves emerge from the solutions to the Schrodinger equation for the hydrogen atom.5... 

As many textbooks correctly report, the number of electrons that can be accommodated into any electron shell coincides with the range of values for the three quantum numbers that characterize the solutions to the Schrodinger equation for the hydrogen atom and the fourth quantum number as first postulated by Pauli. 

For the sake of simplicity, we will here confine ourselves to consider a system of N electrons moving in a given nuclear framework. The stationary states of such a system are described by the solutions to the Schrodinger equation... 

The problem of finding the best approximation of this type and the best one-electron set y2t. . ., y>N is handled in the Hartree-Fock scheme. Of course, a total wave function of the same type as Eq. 11.38 can never be an exact solution to the Schrodinger equation, and the error depends on the fact that the two-electron operator (Eq. 11.39) cannot be exactly replaced by a sum of one-particle operators. Physically we have neglected the effect of the "Coulomb hole" around each electron, but the results in Section II.C(2) show that the main error is connected with the neglect of the Coulomb correlation between electrons with opposite spins. 

In order to get at least a formal definition of the problem, we will write the exact solution to the Schrodinger equation (Eq. II. 1) in the form... 

Solutions to the Schrodinger equation (3.5) are called one-electron wavefunctions or orbitals and take the form in Eq. (3.6)... 

Values such as — 0, = -3, and y are unacceptable because they do not represent solutions to the Schrodinger equation, meaning that they do not correspond to reality. 

The method used to propagate solutions to the Schrodinger equation [61] requires T to be represented on a grid of points distributed in (/ , r, y), which we will denote using the labels klm >. The first term in T is given by... 

Perturbation theory provides a procedure for finding approximate solutions to the Schrodinger equation for a system which differs only slightly from a system for which the solutions are known. The Hamiltonian operator H for the system of interest is given by... 

At this point in the derivation, so as to simplify the notation, the subscript for a particular solution to the Schrodinger equation (2.1) and its associated energy will be dropped. Thus Eq. (2.7) can be rewritten as ... 

The exact form of the wavefunction is also conditioned, however, by the observation that electrons possess spin quantum numbers of either +f of Consequently, physically correct solutions to the Schrodinger equation (2.1) must be antisymmetric. Mathematically, this condition can be written as ... 

If V(R) is known and the matrix elements Hap are evaluated, then solution of Eq. (10) for a given initial wavepacket is the numerically exact solution to the Schrodinger equation. 

Solutions to the Schrodinger equation Hcj) = E(f> are the molecular wave functions 0, that describe the entangled motion of the three particles such that (j) 4> represents the density of protons and electron as a joint probability without any suggestion of structure. Any other molecular problem, irrespective of complexity can also be developed to this point. No further progress is possible unless electronic and nuclear variables are separated via the adiabatic simplification. In the case of Hj that means clamping the nuclei at a distance R apart to generate a Schrodinger equation for electronic motion only, in atomic units,... 

The idea of the LvN method for quantum systems first introduced by Lewis and Riesenfeld (H.R. Lewis et.al., 1969) is to solve Eq. (17) and then find the solution to the Schrodinger equation as an eigenstate of the operator in Eq. (17). In quantum field theory the wave functional to the Schrodinger equation is directly given by the wave functional of the operator... 

The second class of theories can be characterized as attempts to find approximate solutions to the Schrodinger equation of the molecular complex as a whole. Two approaches became important in numerical calculations perturbation theory (PT) and molecular orbital (MO) methods. 

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... 

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