Centre-of-mass wavefunction

Notice that odd n (i.e. even parity V (r)) and odd j (i.e. even centre-of-mass wavefunction) implies and even n (i.e. odd parity V (r)) and odd j implies as the reflection operator reflects both the centre-of-mass and relative coordinates, and hence exchanges the electron and hole. 

Now, the centre-of-mass wavefunctions also form an orthonormal set, ... 

Wavefunctions by themselves can be very beautiful objects, but they do not have any particular physical interpretation. Of more importance is the Bom interpretation of quantum mechanics, which relates the square of a wavefunction to the probability of finding a particle (in this case a particle of reduced mass /r vibrating about the centre of mass) in a certain differential region of space. This probability is given by the square of the wavefunction times dx and so we should concentrate on the square of the wavefunction rather than on the wavefunction itself. 

We need to be clear about the various coordinates, and about the difference between the various vector and scalar quantities. The electron has position vector r from the centre of mass, and the length of the vector is r. The scalar distance between the electron and nucleus A is rp, and the scalar distance between the electron and nucleus B is tb- I will write / ab for the scalar distance between the two nuclei A and B. The position vector for nucleus A is Ra and the position vector for nucleus B is Rb. The wavefunction for the molecule as a whole will therefore depend on the vector quantities r, Ra and Rb-It is an easy step to write down the Hamiltonian operator for the problem... 

This is referred as BO ansatz. This ansatz is taken as a variational trial function. Terms beyond the leading order in m/M are neglected m is the electronic and M is nuclear mass, respectively). The problem with expansion (4) is that functions /(r, R) contain except bound states also continuum function since it includes the centre of mass (COM) motion. Variation principle does not apply to continuum states. To avoid this problem we can separate COM motion. The remaining Hamiltonian for the relative motion of nuclei and electrons has then bound state solution. But there is a problem, because this separation mixes electronic with nuclear coordinates and also there is a question how to define molecule-fixed coordinate system. This is in detail discussed by Sutcliffe [5]. In the recent paper by Kutzelnigg [8] this problem is also discussed and it is shown how to derive adiabatic corrections using, as he called it, the Bom-Handy ansatz. There are few important steps to arrive at formula for a diabatic corrections. Firstly, one separates off COM motion. Secondly, (very important step) one does not specify the relative coordinates (which are to some extent arbitrary). In this way one arrives at relative Hamiltonian Hrd [8] with trial wavefunction If we make BO ansatz... 

The expansion into partial waves is of importance for the desired wavefunction of an emitted electron, because it provides a classification into individual angular momenta ( which refer to the centre of mass of the atom. As a starting point, the expansion of a plane wave will be considered. If a certain origin is selected and the direction of k is chosen to agree with the z-axis (the quantization axis see... 

The interaction between a helium atom and the LiH molecule has been described using a SCVB wavefunction built up using just 25 structures. Interaction energies, computed along different approaches of the two moieties, compare extremely well with a corresponding traditional SCVB calculation using many more structures. Even a very small energy minimum of about 0.01 mHartree is perfectly reproduced for He at a distance of 7 =11 bohr from the centre of mass of the LiH molecule (collinear approach of He to H—Li). 

In the quantum scattering approach the collision is modelled as a plane wave scattering off a force field which will in general not be isotropic. Incident and scattered waves interfere to give an overall steady state wavefunction from which bimolecular reaction cross-sections, cr, can be obtained. The characteristics of the incident wave are determined from the conditions of the collision and in general the reaction cross-section will be a function of the centre of mass collision velocity, u, and such internal quantum numbers that define the states of the colliding fragments, represented here as v and j. Once the reactive cross-sections are known the state specific rate coefficient, can be determined from. 

An instructive description of the H + H2 reaction was provided by McCullough and Wyatt (1971a, b). They constructed a wavepacket and followed its time development on the Porter-Karplus surface. They introduced centre-of-mass coordinates appropriate for reactants and used these for all times. The wavefunction F at time t = n At was constructed, from the time-evolution operator U, in the form... 

Another modified collinear model was developed by Connor and Child (1970). They constrained atoms A, B and C to a line, but allowed this to rotate. In this way impact parameters could be different from zero, although velocities remained always in the same plane. The two coordinates (X, x) in centre of mass were replaced by natural ones, (s, p). The wavefunction F was taken as a product of an adiabatic vibrational state (s, p) times a translational function i6), with 0 the axial angle. Then... 

These atoms consist of an electron and a nucleus, both of which are in motion. Since wave-like properties have to be associated with both particles, the full wave equation for the atom involves a total of six variables, and such equations are usually difficult to solve. Fortunately, the motion of the atom as a whole can be separated into two parts (a) the translational motion of the centre of mass, for which particle-in-a-box wavefunctions are appropriate, and (b) the motion of the nucleus and electron relative to the centre of mass (see Figure 6.2a). 

Notice that two quantum numbers specify the exciton eigenstates, eqn (6.13) or eqn (6.16) the principle quantum number, n, and the (pseudo) momentum quantum number, K (or fUj). For every n there are a family of excitons with different centre-of-mass momenta, and hence different centre-of-mass kinetic energy. Odd and even values of n correspond to the relative wavefunction, tl>n r), being even or odd under a reversal of the relative coordinate, respectively. We refer to even and odd parity excitons as excitons whose relative wavefunction is even or odd imder a reversal of the relative coordinate. This does not mean that the overall parity of the eigenstate (eqn (6.12)), determined by both the centre-of-mass and relative wavefiictions, is even or odd. The number of nodes in the exciton wavefunction, V rt( ), is n— 1. Figure 6.2 illustrates the wavefunctions and energies of excitons in the effective-particle model. 

We note that the effective-particle description is still valid when there is selftrapping. In this case the centre-of-mass wavefuctions are not the particle-in-the-box wavefunctions appropriate for a linear chains (eqn (9.130)), but they are the ortho-normalized functions appropriate for the particular potential well trapping the effective particle. The key point is that because these are ortho-normalized functions and because the potential wells for the excitons and polarons are very similar the selection rules for interconversion are still valid. Thus, interconversion occurs between a pair of states with the same center-of-mass quantum numbers. 

We shall be concerned mainly with the wavefunctions of electrons in atoms and in this case the predominant contribution to the potential comes from the Coulomb attraction of the nucleus. This potential is. spherically symmetric and therefore V(r) is a function of tlie radial coordinate r alone, This enables the Schrodinger equation to be separated into three differential equations which involve r, 0, and (j> separately. If we consider the motion of a single electron of mass m about a nucleus of mass M we can separate off the centre-of-mass motion and consider only the relative motion of the electron. In spherical polar coordinates equation C3.8) becomes (Problem 3.3)... 

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