Schroedinger equation approximate solutions

While this particular problem of mechanical stability can be solved in principle by means of the Schroedinger equation, exact solutions have never been obtained for any polyatomic molecules except H2 and Hf. Later on we shall consider some of the approximate treatments which have been made. 

We cannot solve the Schroedinger equation in closed fomi for most systems. We have exact solutions for the energy E and the wave function (1/ for only a few of the simplest systems. In the general case, we must accept approximate solutions. The picture is not bleak, however, because approximate solutions are getting systematically better under the impact of contemporary advances in computer hardware and software. We may anticipate an exciting future in this fast-paced field. 

In PPP-SCF calculations, we make the Bom-Oppenheimer, a-rr separation, and single-electron approximations just as we did in Huckel theor y (see section on approximate solutions in Chapter 6) but we take into account mutual electrostatic repulsion of n electrons, which was not done in Huckel theory. We write the modified Schroedinger equation in a form similar to Eq. 6.2.6... 

By systematically applying a series of corrections to approximate solutions of the Schroedinger equation the Pople group has anived at a family of computational protocols that include an early method Gl, more recent methods, G2 and G3, and their variants by which one can anive at themiochemical energies and enthalpies of formation, Af and that rival exper imental accuracy. The important thing... 

Fig. 35). The potential energy curves and the transition dipole moment are taken from [117]. The time evolution of the populations on the ground and excited states is shown in Fig. 36 More than 86% of the initial state is excited to the B state within the period shorter than a few femtoseconds. The integrated total transition probability V given by Eq. (173) is P = 0.879, which is in good agreement with the value 0.864 obtained by numerical solution of the original coupled Schroedinger equations. This means that the population deviation from 100% is not due to the approximation, but comes from the intrinsic reason, that is, from the spread of the wavepacket. Note that the LiH molecule is one of the...

Equations A.5, A.9, and A.10 define the BO approximation. Note that two approximations (neglect of the second term of the left-hand side of eq. A.7 and off-diagonal terms in A.6) were made in order to obtain this result. Therefore, we see that the BO expression A.10 is not an exact solution of the total Schroedinger equation A.l. 

Abstract. Calculations of the non-linear wave functions of electrons in single wall carbon nanotubes have been carried out by the quantum field theory method namely the second quantization method. Hubbard model of electron states in carbon nanotubes has been used. Based on Heisenberg equation for second quantization operators and the continual approximation the non-linear equations like non-linear Schroedinger equations have been obtained. Runge-Kutt method of the solution of non-linear equations has been used. Numerical results of the equation solutions have been represented as function graphics and phase portraits. The main conclusions and possible applications of non-linear wave functions have been discussed. 

Solution of the Schroedinger equation, H p = Ef, appropriate to this problem has only been accomplished by means of successive perturbation calculations. The zero-order approximation is a spherical approximation in which a given outer electron is assumed to move in the average potential of the other outer electrons as well as of the core electrons. Then the free-ion Hamiltonian becomes... 

The Bom-Oppenheimer approximation is not always correct, especially with light nuclei and/or at finite temperature. Under these circumstances, the electronic distribution might be less well described by the solution of a Schroedinger equation. Non-adiabatic effects can be significant in dynamics and chemical reactions. Usually, however, non-adiabatic corrections are small for equilibrium systems at ordinary temperature. As a consequence, it is generally assumed that nuclear dynamics can be treated classically, with motions driven by Bom-Oppenheimer potential energy functions ... 

The Schroedinger equation cannot be solved exactly except for very simple systems like the hydrogen atom. For molecules, we must be satisfied with an approximate solution of H V(r) = is F(r). In recent years, owing to the work of Pople, Gordon, and others, agreement between MO approximations and such experimental results as exist has been brought to a level that makes quantum thermochemistry competitive with experimental thermochemistry in reliability. 

As a first step in obtaining an approximate solution to the molecular Schroedinger equation, we agree to regard the many electron wave function P(r) as having been broken up into molecular orbitals y/i... 

Each time the term solution is used in reference to the Schroedinger equation from this point on, the reader should assume that the solution is approximate. According to the variational principle, one varies the j/i so as to obtain a minimum but approximate E which is an upper bound of the true energy. 

In general, the development of approximation methods for the solution of the many-electron Schroedinger equation is a challenge for physicists because no exact numerical solutions can be found apart from very few cases of a small number of electrons, such as the helium atom. The main difficulty arises because of the electron-electron interaction, which is a two-particle operator. Thus, increasing the accuracy of solutions implies increasing the computer time needed for the numerical calculations, and the cost becomes prohibitive even for molecules with a few atoms. 

Thus we have formally, and exactly, converted the master equation to a Schroedinger equation. This has the substantial advantage that we can apply well-known approximations in quantum mechanics to obtain solutions to the master equation. In particular we refer to the W.K.B. approximation valid for semiclassical cases, those for which Planck s constant formally approaches zero. The equivalent limit for (3.8) is that of large volumes (large munbers of particles). Hence we seek a stationary solution of (3.8), that is the time derivative of Px X,t) is set to zero, of the form... 

This total energy fo (with the fixed positions of atoms) provides a potential for nuclear motion. Such quantity is called Potential Energy Function (PES) and in molecular dynamics simulations is approximated by a force field. In terms of quantum mechanics solutions to the nuclear Schroedinger equation ... 

In the case of the vibration/rotation of a diatomic molecule, the /(/ -t 1) term in the radial Schroedinger equation is approximated via a power series expansion (see Equation 6-18). This approximation is sufficient for vibration/rotation of diatomic molecules because the distance of separation of the two nuclei does not vary greatly between rotational states. In the case of electronic states however, the separation of the electron and the nucleus varies widely between states and a power series expansion is inappropriate. Eortunately, the solution to Equation 8-6 is well known. There are an infinite number of solutions for each value of / and each one is designated by a quantum number n. Each state is called an atomic orbital (AO). The quantum numbers that distinguish the possible states is given as follows. 

Hamiltonian models are classified according to then-level of approximation. The features of Schroedinger (S), Born-Oppenheimer (BO), and McMillan-Mayer (MM) level Hamiltonian models are exemplified in Table I by a solution of NaCl in H2O. The majority of investigations on electrolyte solutions are carried out at the MM level. BO-Level calculations are a precious tool for Monte Carlo and molecular dynamics simulations as well as for integral equation approaches. However, their importance is widely limited to stractural investigations. They, as well as the S-level models, have not yet obtained importance in electrochemical engineering. S-Level quantum-mechanical calculations mainly follow the Car-Parinello ab initio molecular dynamics method. 

---