Computational biochemistry ferenc Bogár György Ferency
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- 3.3. The potential energy surface
- 4. Solving the Schrödinger equation of the stationary N-electron system
Ra,rj) is an eigenfunction of Ĥelec with Ek eigenvalue. Acting  onto the χk(Ra) Ψk(Ra,rj) product – without the details of the calculation – we get for the first term on the right side in Eq. 11.6
where  the new operator contains the first and second derivatives with respect to the variables of nuclei. Let’s substitute Eq. 11.7 into Eq. 11.6 then multiply both side with Ψ*l(Ra,rj) (the star means the complex conjugation for complex value wavefunction) and integrate according to the variables of the electron. This way we can get matrix element of an operator (e.g.: ) with respect to the chosen basis set (in our case Ψk(Ra,rj)). Hereafter the (l,k)-th matrix element of an operator will be denoted by the indexes in the upper-right position of the operator, so
By applying the orthonormality of the basis functions Ψk(Ra,rj), the form of Eq. 11.6 turns into
with the help of Eq. 11.7 and Eq. 11.8. In the system of Eq. 11.9 the χl(Ra) functions and the E are the unknown quantities and the equations form a coupled system of equations because of the last term in Eq. 11.9. The El quantities can be determined by solving the eigenvalue equation in Eq. 11.5. If we set the k ≠ l matrix elements of to zero, then we decouple the equations in Eq. 11.9 so we will have Ma independent equations. This approximation, when the kl elements are set to zero called as adiabatic approximation, and it should be distinguished from the Born-Oppenheimer approximation, where the whole matrix is set to zero. It means that in the Born-Oppenheimer approximation both the diagonal and the off-diagonal elements of the operator are set to zero. Thus the principle question is now the solvation of eigenvalue equation in Eq. 11.5. Before we start to focus onto this problem, we would like to have a few words on the consequences of the Born-Oppenheimer approximation.
3.3. The potential energy surface Let’s write Eq. 11.9 in the Born-Oppenheimer approximation as
The form of the equation is strongly reminds of the form of a Hamilton-operator, where the nuclei are handled independently from the electrons and they move in the “potential” denoted by El . Therefore the elimination of the term with the operator really means the full decoupling of the motion of nuclei from the electron system. Moreover, the El depends implicitly on the geometry of the nuclei, so changing the geometry of the nuclei (i.e. moving them) provide different El values. Plotting the El values as the function of the geometrical parameters, we can get a “surface” in the parametrical coordinate system. This surface called as potential energy surface (PES). Taking into account the principle of minimum total potential energy, the nuclei prefer those geometries where the El surface has its minima. This is the theoretical basement of geometrical optimization calculations, where the total energy of the electron system calculated in different arrangement of the nuclei. It is worth to note that the state of the electron system (e.g. ground state or any excited state) cannot change on a surface: different states of the electrons provide different PESs.
As an example, we present the ground state PES of the ethanol molecule with respect to two dihedral parameters (see figure 11-1). The two parameters describe the rotation of the methyl and the hydroxyl groups around the C-C and the C-O bonds, respectively. The ground state energies of the electron system are calculated systematically in 30° steps. The different minima and maxima of the surface can be easily interpreted by staggered and eclipse geometries of the rotating groups. 
Figure 11.1. The PES of the ethanol regarding the systematic rotation of the methyl and the hydroxyl group. We would like to note that in a later chapter a case study will show the application of PES in a conformational analysis of a biologically interesting molecule. 4. Solving the Schrödinger equation of the stationary N-electron system Let’s have a quick summary about the approximations which have been applied until this point in the QM level discussion of atoms and molecules. First, non-relativistic situation has been chosen since we use the Schrödinger equation. Then we separated the time coordinate and considering only the stationary states following by the separation of the nucleus and the electron coordinates with the help of the adiabatic and the Born-Oppenheimer approximation. To determine the ground or excited states energies (and wave functions) of the electron system we have to solve the eigenvalue equation of the  operator which is still not an easy task. It is well known that system with more than 2 electrons cannot be solved analytically, and one can imagine that biologically or chemically interesting systems easily contains several hundreds of electrons. So, further approximations or omissions are still surely necessary.
Before having the next step, we have to clarify an important fact regarding identical particles. Two particles are considered as identical, if all of their inner properties (mass, charge, spin, etc.) are the same. In classical mechanics it is not problematic to distinguish two identical particles since both of them have a well determined path. However, this is not the case in quantum mechanics. If we have a QM system with identical particles they are indistinguishable and this requires the introduction of a new postulate. Namely, we require that the wave function of a system must be completely symmetric if it is built up from identical particles with integer spin (bosons). On the other hand, the wave function of a system must be completely antisymmetric if it contains identical particles with half spin (fermions). The antisymmetric property means that the wave function of the system must change its sign for the exchange of two particles. Since the electron has half spin therefore the wave function of the electron system must be totally antisymmetric. Until this point we did not specify the concrete form of the potential term in the Ĥelec operator. If we do not have any additional external fields (e.g. extra electrostatic or magnetic field) then the  potential contains three terms based on electrostatic interactions: the repulsion of the nuclei, the attraction of the nuclei and the electrons and the repulsion of the electrons. More concretely,
where Za means the total charge of the nucleus in the position of Ra. It is important to note that the kinetic energy operator of the electron system has the form
Taking these forms, the terms in the Ĥelec operator can be classify according to the sub-terms in the summations in Eq. 11.11 and Eq. 11.12. The classification is based on the number of electrons appear through variables in a sub-term. Regarding the potential operator, the first term does not contain any electron variable, it contains the coordinates of one electron in each sub-term and the last term contains the coordinates of two particles in every sub-terms. The kinetic energy operator contains the variable of one particle in each sub-terms in Eq. 11.12. Aside from the nucleus-nucleus interaction term, we have two classes: the one-particle and the two-particle operators. Thus, the kinetic and the nucleus-electron interaction operators are one-particle operators, while the electron-electron interaction operator is a two-particle operator. The nucleus-nucleus interaction operator behaves as a constant multiplier regarding the wave function of the electron system and in the eigenvalue-equation it shifts with a constant the Ej values so it will be taken into account only at the end of the calculations. Focusing onto the eigenvalue equation of the electrons, such methods will be reviewed first briefly which replace the two-particle operator in the original potential with an effective one-particle potential. This way the electrons will move in a potential created by the nuclei and the new effective potential without having interaction with each other. It can be also interpreted as every electron in the system would move independently, and the effect of the other electrons would be taken into account in an averaged manner by the effective potential. That’s why these approaches usually called as independent particle approximation or mean-field approximation. The primary consequence of these methods is that the original equation with 3Ne (or by taking into account the spin, with 4 Ne) variables is substituted by Ne equations with 3 (or 4) variables. So, the Hamilton operator of the electron system without the nucleus-nucleus interaction can be transformed into the sum of one-particle operators. Hence we can write
but the explicit form of  is still unknown. If we can determine the term then the variables can be separated in the eigenfunction by applying a product form. Here the terms in the product are the one-particle functions which depends on the 3 (or 4) variables of one electron. Moreover, this is the first step to include the orbital picture into the QM level treatment of the electron system. However we cannot forget about the antisymmetric property of the wave function regarding the electron system. Therefore we cannot build up the total N-particle wave function as a simple product of the one-particle functions, since it must satisfy the antisymmetric requirement. Considering the simplest two-particles system, the form
is the simplest choice for an antisymmetric expression, where the factor in the beginning of the expression is the consequence of the normalization. One can easily check that if we exchange the two variables, then the expression change the sign. Similarly, in case of three electrons a 6-members product provide the same results, as well as a 24-members product in case of four particles. John C. Slater was the first, who derived the antisymmetrical total wave function with the help of matrix determinant. Namely, he wrote the total wave function as the result of the calculation of the determinant of the following NexNe matrix
The vertical lines refer to the determinant calculation, and one can easily check that this is the general and compact form of the previously mentioned 2, 3 or 4 particles cases. It is worth to note that the determinant form is not the only one which can provide a totally antisymmetric wave function based on the product of one-particle functions. However this is one of the simplest ones, and the question now is only that, how can we determine the unknown effective potential? The firs answer for this problem was given by the Hartree-Fock method.
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