Schrodinger equation trial wavefunction

Now we need to determine the values of the constants c . Recall that this equation was determined by substituting a trial wavefunction into the Schrodinger equation, so that if the harmonic oscillator system has wavefunctions that are eigenfunctions of the Schrodinger equation, those wavefunctions would be of the form given in equation 11.8 [that is, R = f(x)]. By identifying the constants, we... 

So far the form of 4, has not been specified. While the Schrodinger equation defines clear conditions for valid state functions of the system, it does not offer a practical method to actually obtain the exact solution VP or vPel. As a consequence, iterative procedures have been developed that start with some trial wavefunction, and subsequently improve their quality until some predefined convergence criterion is fulfilled. 

The rationale behind this approach is the variational principle. This principle states that for an arbitrary, well-behaved function of the coordinates of the system (e.g., the coordinates of all electrons in case of the electronic Schrodinger equation) that is in accord with its boundary conditions (e.g., molecular dimension, time-independent state, etc.), the expectation value of its energy is an upper bound to the respective energy of the true (but possibly unkown) wavefunction. As such, the variational principle provides a simple and powerful criterion for evaluating the quality of trial wavefunctions the lower the energetic expectation value, the better the associated wavefunction. 

Even when confining the variation of the trial wavefunction to the LCAO-MO coefficients c U, the respective approximate solution of the Schrodinger equation is still quite complex and may be computationally very demanding. The major reason is that the third term of the electronic Hamiltonian, Hel (Equation 6.12), the electron-electron repulsion, depends on the coordinates of two electrons at a time, and thus cannot be broken down into a sum of one-electron functions. This contrasts with both the kinetic energy and the electron-nucleus attraction, each of which are functions of the coordinates of single electrons (and thus are written as sums of n one-electron terms). At the same time, orbitals are one-electron functions, and molecular orbitals can be more easily generated as eigenfunctions of an operator that can also be separated into one-electron terms. 

For an atom the evaluation of the potentials in Eq. (2) is achieved straightforwardly, using trial wavefunctions, by integrating the charge density radially. Once the potentials are known the wavefunctions are obtained by numerical integration of the radial part of the Schrodinger equation, and the process is continued iteratively to self-consistency. In molecules the loss of spherical symmetry makes the procedure much more difficult, and in particular for molecules containing heavy atoms the number of matrix elements which must be evaluated becomes prohibitive. Calculations on the uranyl ion have employed three different approaches to circumvent this difficulty. 

The antisymmetry feature of the trial wavefunction is required because the Schrodinger equation does not exclude those trial functions that do not fulfill the antisymmetry requirement. Therefore, not all solutions to eq. (1) are acceptable the search for a solution must be restricted to those trial wavefunctions that are antisymmetric with respect to the exchange of any two electrons. This implies that the basis set should have a form that allows the wavefunction to be antisymmetric to the interchange of any two electrons. As a result, the Vee energy has two parts a direct term,... 

Cases of three or more electrons were very difficult to treat by the above methods. For instance, for three-electron systems, it is required to have six terms in the expansion of each basis function in order to comply with the antisymmetry criterion, and each term must have factors containing ri2, ri3, r23, etc., if we want to accelerate the convergence. There is, indeed, a real problem with the size of each trial wave function. A symmetrical wavefunction requires that the trial basis set for helium contain two terms to guarantee the permutation of electrons. For an N-electron system, this number grows as N . For a ten-electron system like water, it would be required that each basis set member have more than 3 million terms, and this is in addition to the dependence on 3N variables of each of the terms. These conditions make the Schrodinger equation intractable for systems of even a few electrons. Just the bookkeeping of these terms is practically impossible. 

Undoubtedly, the methods most widely used to solve the Schrodinger equation are those based on the approach originally proposed by Hartree [1] and Fock [2]. Hartree-Fock (HF) theory is the simplest of the ab initio or "first principles" quantum chemical theories, which are obtained directly from the Schrodinger equation without incorporating any empirical considerations. In the HF approximation, the n-electron wavefunction is built from a set of n independent one-electron spin orbitals which contain both spatial and spin components. The HF trial wavefunction is taken as a single Slater determinant of spin orbitals. 

In the diffusion QMC (DMC) method [114. 119]. the evolution of a trial wavefunction (typically wavefunctions of the Slater-Jastrow type, for example, obtained by VMC) proceeds in imaginary time, x = it, according to the time-dependent Schrodinger equation, which then becomes a diffusion equation. All... 

The procedure is called the linear combination of atomic orbitals (LC AO) approximation and can be used for molecules of any size. H2 is a special case in that a wavefunction can be found that will solve the Schrodinger equation exactly, yet the MO approach will be used so that molecular orbitals can be derived. The simplest trial function for the H2+ system is written ... 

However, it is the Pauli principle which prevents us from simply ignoring the existence of electron spin altogether. The trial wavefunction must be antisymmetric with respect to the exchange of the coordinates (space-spin) of any two particles. Without this constraint the solutions of the many-electron Schrodinger equation would be wrong there are many more solutions of the Schrodinger equation than there aje antisymmetric solutions of that equation. Electron spin, at this level, simply ensures that the spatial part of the wavefunction behaves properly when the electrons coordinates are permuted. Thus, notwithstanding the manipulational convenience of the use of spin functions it would be attractive to be able to deal explicitly only with a spatial trial function and solve a spatial variational problem. 

We take a variational approach so that there is no question of requiring an exact solution of the Schrodinger equation for reference. Let J be a variational trial function for the valence electrons of a many-electron system and let h be the valence many-electron Hamiltonian. We seek a minimum in the mean value of H with respect to such (normalised) trial functions together with the constraint that be orthogonal to the wavefunction of a subset of the electrons (the core). We will then recast the equation into a pseudopotential form and examine this form with a view to modelling the pseudopotential. 

Table 1 illustrates the six wavefunctions that are available in ATOMPLUS along with sample output from solving the Schrodinger equation with these trial wavefunctions for the He atom. Table 1 depicts only the spatial part of the wavefunction. Each spatial wavefunction is symmetric the total wavefunction is the product of this spatial part and the spin part which is antisymmetric. 

Q Show that the trial function y/ - Ne, where N and k are constants, is a solution of the Schrodinger equation for the hydrogen atom when the constant k has a particular value. Hence, calculate the energy associated with this wavefunction. 

Although it is possible to obtain exact solutions to the simplified Schrodinger equation given in equations (8.8) and (8.9), the resulting wavefunction is complicated, and it provides little insight into the wave-fimctions that might be used for other diatomic molecules. For this reason it wiU be more instructive to examine trial wavefunctions that have been constructed by a linear combination of hydrogen Is atomic orbitals ... 

When the Schrodinger equation was solved for the H atom, we obtained an infinite number of solutions I s, 2s, 2p, 3s, 3p, 3d, and so on. Although it would appear from this example that there are only two MOs for the H2 ion, there are in fact an infinite number of MOs. When applying the LCAO-MO method, we could have included higher order terms in our trial wavefunction, as shown in Equation (10.28). 

The computational problem of the CC method is determination of the cluster ampUtudes t for all of the operators included in the particular approximation. In the standard implementation, this task follows the usual procedure of left-multiplying the Schrodinger equation by trial wavefunctions expressed as determinants of the HF... 

The operators have now been dealt with and the wavefunction will cancel on both sides. This shows that we have found the value for jS that makes our trial function into a working solution for the harmonic oscillator Schrodinger equation. 

The quantum numbers v and J stand collectively for the quantum numbers of all the vibrational modes and for the quantum numbers of the rotational motion of the whole molecule. When we insert this approximate trial wavefunction in the time-independent Schrodinger equation, Eq. (2.5), neglect two small terms [see Exercise 2.21 and make use of the time-independent field free electronic Schrodinger equation, Eq. (2.10), we obtain the nuclear Schrodinger equation... 

The reason the Schrodinger equation could be solved exactly for the hydrogen atom was that the potential V was a simple function of r oiUy that describes the attraction between the single electron and the nucleus. In a system with many electrons, the potential function would have to include the repulsive interaction between an electron and all the other electrons. This is an extremely difficult calculation which can only be carried out numerically using an iterative scheme. A guess is made for the wavefunction of each electron from the hydrogen wavefunction. Then the trial wavefunctions are adjusted... 

A special and powerful use of variation theory is with linear variational parameters. That means that the trial wavefunction is taken to be a linear combination of fimctions in some chosen set. The adjustable parameters are the expansion coefficients of each of these functions. This is, of course, a specialization of the way in which variation theory can be used, but it is powerful because the resulting equations take the form of matrix expressions. Solving the Schrodinger equation becomes a problem in linear algebra, and such problems are ideally suited to computer solution. 

The cluster energies are obtained from the solution of the non-relativistic SchrOdinger equation for each system. The expansion of the trial many-electron wavefunction delineates the level of theory (description of electron correlation), whereas the description of the constituent one-electron orbitals is associated with the choice of the orbital basis set. A recent review [54] outlines a path, which is based on hierarchical approaches in this double expansion in order to ensure convergence of both the correlation and basis set problems. It also describes the application of these hierarchical approaches to various chemical systems that are associated with very diverse bonding characteristics, such as covalent bonds, hydrogen bonds and weakly bound clusters. 

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