Computational biochemistry ferenc Bogár György Ferency
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If the potential does not depend explicitly on time - so it has the form V(Ra,rj ,t) - then we can look for the wave function solution of the Schrödinger equation in θ(Ra,rj ,t)=Φ(Ra,rj ,t)χ(t) form. This method is called as separation of variables in mathematics since the original many variable function is written as the product of functions with fewer variables than the original one. The new functions still can have many variables but cannot have the same one simultaneously. The advantage of this method is that the original equation with large number of variables is separated into more than one independent equation but with fewer variables. In the present case the original wave function separated into a purely time-dependent function and a function with all the other variables. Substituting the new form into Eq. 11.1 and take the advantage of that the purely time dependent function behaves as a constant function for the kinetic energy operator, after some algebra we can have
The left side of Eq. 11.2 depends only on the time variable while the other side on all the other ones, which can only happen if both sides are constant. Let’s denote this constant first with ω (or in non-atomic units with ħω) and we get χ(t)=A·exp(-i·ωt) by solving the equation from the left side. This way the general form of the purely time-dependent part of the wave function has been determined. Regarding the right-side of Eq. 11.2 , if Φ(Ra,rj) is moved from the denominator next to the ω constant, and the constant will be denoted hereafter as E (which is understandable if we take into account the non-atomic units form of the constant ħω), then we can get the Schrödinger equation of stationary states, namely
The newly defined operator Ĥ is called as the Hamilton-operator of the system, and E is the total energy of the system. Since Eq. 11.3 does not depend explicitly on time therefore Φ(Ra,rj) will not change in time as well. These solutions are known as stationary states, and the next step is the solution of this equation. It is worth to mention that those equations where we try to find those vectors, on which the effect of an operator is at most a scalar multiplication, are known as eigenvalue equations. The solutions of such an equation are called eigenfunctions (or eigenvectors), and the scalars are the eigenvalues. This is the case in Eq. 11.3 if one looks the left and right parts of the formula. The Ĥ operator acts on the Φ(Ra,rj) function and provide its scalar multiplied form (E·Φ(Ra,rj)). 3.2. The adiabatic and the Born-Oppenheimer approximations The determination of the stationary states can be achieved by the solution of Eq. 11.3, however, it is still a complicated problem because of the large number of variables. Thus the primary aim of the further approximations is still to reduce the number of variables. Taking into account that from biological and chemical point of view the most important effects (bond breaking or creation) are related to the electron system, we try to simplify Eq. 11.3 that way that the variables of nuclei will be eliminated. Since the total mass of the nuclei is 3-4 orders of magnitude larger than the total mass of the electron system, the electrons can instantaneously follow the motion of nuclei. If we ignore the backward coupling of the electron system to the nuclei, the motion of nuclei and the motion of electrons are considered independently. Thus the coordinates of the nuclei can be considered as parameters for the electron coordinate in a chosen instant. Fixing the geometry of nuclei we can get the eigenfunction of the electron system related to the chosen nucleus-geometry. Introducing the notation Ψk(Ra,rj) and Ek for the k-th eigenfunction and eigenvalue of the electron system in the fixed nuclei, where the lowest eigenvalue belongs to k=1. It can be shown that the Ψk(Ra,rj) eigenfunctions form a basis in the vector-space of all solutions related to any fixed geometry. Mathematically it means
The χk(Rk) coefficients certainly depends on the fixed geometry of the nuclei, therefore they contain the nuclei coordinates as parameters. If we use the decomposition of the  operator based on the separation of the nuclei and electron coordinates ( ) and introduce the Eq. 11.5
called as the Hamilton operator of the electron system, then the form of Eq. 11.3 will change to
by substituting Eq. 11.4 and Eq. 11.5 into Eq. 11.3. Here we applied the property of addition of operators then we also used that the Ψk( Directory: tartalom -> tamop412A tamop412A -> Agricultural Economics II. Popp, József Agricultural Economics II tamop412A -> Ethology practical Vilmos Altbäcker Márta Gácsi András Kosztolányi Ákos Pogány Gabriella Lakatos tamop412A -> Operating Systems Lecture Notes Attila Dr. Adamkó Operating Systems Lecture Notes tamop412A -> Agroinformatics Herdon, Miklós Agroinformatics tamop412A -> Paternoster of Programmers Reloaded tamop412A -> Lőrincz, András Mészáros, Tamás Pataki, Béla Embedded Intelligent Systems tamop412A -> This course is realized as a part of the TÁmop 1 A/1-11/1-2011-0038 project tamop412A -> Fundamentals of geology I. (lithosphere) 1 1. The formation of the Earth 1 Download 2.94 Mb. Share with your friends:
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