Vibrational Schrbdinger equation

The description of PCET reactions is particularly challenging due to the quantum mechanical behavior of the ET electrons, the PT electrons, and the transferring protons. The adiabatic mixed electronic/proton vibrational states are calculated when the following Schrbdinger equation is solved for fixed solvent coordinates... 

We note first that the masses of the nuclei are much greater than those of the electrons, Mproton = 1836 atomic units compared to electron = 1 atomic unit. Therefore nuclear kinetic energies will be negligibly small compared to those of the electrons. Typically, the amplitude of nuclear vibration is of the order of 1 % the spread of an electron s probability distribution. In accordance with the Bom-Oppenheimer approximation, we can first consider the electronic Schrbdinger equation... 

The rigid rotor model assumes that the intemuclear distance Risa constant. This is not a bad approximation, since the amplitude of vibration is generally of the order of 1% of i . The Schrbdinger equation for nuclear motion then involves the three-dimensional angular-momentum operator, written J rather than L when it refers to molecular rotation. The solutions to this equation are already known, and we can write... 

There are two possible approaches to solution of the Schrbdinger equation with the model Hamiltonian (9.3). The first is based on direct solution using grid [17] and basis [18] methods or the so-called multiconfiguration time-dependent Har-tree method [19]. (The latter method is a combination of grid and basis set methods in the sense that the time-dependent basis functions are represented on a suitable grid). The second approach uses a Bom-Oppenheimer type separation between the motions of the light and heavy nuclei (the method of adiabatic separation between vibrational variables) [20]. In the second approach the wave function F(s,R) is written as ... 

Next, we consider the solution of the Schrbdinger equation of vibrational motion with the anharmonic PESs... 

In this section we will increase our quantum-mechanical repertoire by solving the Schrbdinger equation for the one-dimensional harmonic oscillator. This systmn is important as a model for molecular vibrations. 

The time evolution of a vibrational wavepacket V (f) is governed by the time-dependent, nuclear Schrbdinger equation ... 

In the previous chapter, we solved the problem of the quantized harmonic oscillator and derived key concepts such as the reduced mass and the isotope shift. We were on the verge of treating rotation but you will soon see it is a two-dimensional problem, which needs to be split into two onedimensional problems. Basically the motion of a gas-phase molecule is translation and free rotation and it takes two coordinates (9, c])) to describe such rotational motion even when we assume constant bond lengths within the molecule. We know from the previous chapter that molecules do vibrate but the motion of the vibrations is much smaller than rotations described by (0, < )). Therefore it is a good approximation to assume constant bond lengths. Thus, we have to solve the Schrbdinger equation for a problem in more than one dimension. 

We shall proceed as follows. We shall first diagonalize the Schrbdinger problem [Eq. (3.46)] with respect to the vibrational and rotational quantum numbers (Section 5.1). We arrive in this way at a Schrodinger equation in the variable p with an effective potential function for each vibration—rotation state. A least squares procedure that includes the numerical integration of the Schrodinger equation for this effective Hamiltonian will be used to determine the harmonic force field and the doubleminimum inversion potential function for ( NHa, NHs), ( ND3, NTa) and NH2D, ND2H (Section 5.2). 

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