Bound states solutions

On pages 133-136 of Pauling and Wilson are tabulated the bound-state solutions of this problem for n ranging up to n=6 and 1-values up to 1=5. 

At this point it is interesting to discuss the solutions of the DBC Hamiltonian briefly. As Brown and Ravenhall pointed out in 1951 [20], the DBC Hamiltonian does not have bound state solutions. To illustrate this point, let us consider the stationary state DBC Hamiltonian of two non-interacting fermions in an external potential given by... 

If the kinetic balance condition (5) is fulfilled then the spectrum of the L6vy-Leblond (and Schrodinger) equation is bounded from below. Then, in each case there exists the lowest value of E referred to as the ground state. In effect, this equation may be solved using the variational principle without any restrictions. On the contrary, the spectrum of the Dirac equation is unbounded from below. It contains the negative ( positronic ) continuum. Therefore the variational principle applied unconditionally would lead to the so called variational collapse [2,3,7]. The variational collapse maybe avoided by properly selecting the trial functions so that they fulfil the boundary conditions specific for the bound-state solutions [1]. 

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... 

As a consequence of the attractive region of the interaction, several bound state solutions of the radial Schrodinger equation generally exist whose wavefunctions are localized in the well region. The bound state eigenenergies are negative, discrete and will be subscripted with the vibrational and rotational quantum numbers, v and the normalization... 

To find bound state solutions, W < 0, for the H atom we apply the r = 0 and r = oo boundary conditions. Specifically we require that xp be finite as r — 0 and that xf>—> 0 as r— °°. We can see from Eqs. (2.12) that only the/functions are allowed due to the r — 0 boundary condition. As < we require that ip — 0, and, as indicated by the asymptotic form of the / function, this requirement is equivalent to requiring that sin nv be zero or that v be an integer. Combining the angular function of Eq. (2.7) with the / radial function yields the bound H wavefunction... 

If k is purely imaginary and positive, then these states correspond to bound states with asymptotic behavior L) solutions with positive imaginary values of the wave vector [31]. 

Incidentally we can limit ourselves to real energies, identifying, in the limit to the real axis and the unique bound state solution alternatively the two branches in the continuum, while if the energy is complex we must keep in mind that/+ (k, r) correspond to a unique square integrable solutions if I (k) > 0 and/ (k, r) if I(k) < 0, where I(k) denotes the imaginary part of k. In the case of logarithmic derivative obtains two branches corresponding to the limits e -> 0... 

The eigenvalue problem of the Hamiltonian operator (1) is defined in an infinite-dimensional Hilbert space Q and may be solved directly only for very few simple models. In order to find its bound-state solutions with energies not too distant from the ground-state it is reduced to the corresponding eigenvalue problem of a matrix representing H in a properly constructed finite-dimensional model space, a subspace of Q. Usually the model space is chosen to be spanned by TV-electron antisymmetrized and spin-adapted products of orthonormal spinorbitals. In such a case it is known as the full configuration interaction (FCI) space [8, 15]. The model space Hk N, K, S, M) may be defined as the antisymmetric part of the TV-fold tensorial product of a one-electron space... 

Positive pair-binding energies, and hence the effective attractive interaction between electrons, arc necessarily a core polarization effect. If all of the 60 valence electrons were treated as an inert Fermi sea, the net interaction between two electrons added to a given molecule would necessarily be repulsive there is no bound-state solution to the Cooper problem with purely repttlsive interactions. It is the dynamic interactions with the valence electrons that are crucial in producing overscreening of the purely bare repulsive interaction. Although the first-order term does not involve any virtual excitations, the second-order theory includes important core-polarization effects. 

Here E, are the Dirac eigenvalues, index s runs over the complete spectrum of the Dirac Hamiltonian. This spectrum includes the solutions with positive and negative energy values E,. For the bound-state solutions the... 

The practical question remains how can we find bound-state solutions of equations based on the Dirac-Coulomb-Breit operator The naive approach to the solution of a He-like problem would be to assume that the two-electron wavefunction, can be written as a complete-set expansion of the form... 

When the interaction potential V r) is attractive, the Schrodinger equation (1.15) may support bound-state solutions at energies E <0 (below the dissociation energy) as well as scattering solutions at energies > 0, as shown in Figure 1.1. In practice, almost all systems of interest support many bound states. 

Numerical methods for finding bound-state solutions of Equation 1.15 are described in Section 1.3.1.3. However, in conceptual terms a considerable amount may be understood in terms of semiclassical arguments [7]. In semiclassical methods, the Schrodinger equation is expanded semiclassically in powers of h. The resulting first-order JWKB (Jeffreys-Wentzel-Kramers-Brillouin) quantization condition gives remarkably accurate results for the vibration-rotation energies E l of diatomic molecules ... 

This expansion is known as the Born-Oppenheimer expansion. Formally, Eq. (5) is exact, since the set j(r, R) is complete. It is only when the expansion is truncated that approximations are introduced. The Born-Oppenheimer expansion certainly provides a perfectly valid ansatz if f(r, R) describes a bound state solution of the full Schrodinger equation... 

We have bound state solutions fa,( ), when Eg is less than energy Vo(Q of Iho lowest dissociation channel of the... 

For E > 0 (unbound states), solutions exist for all values of E. For E < 0 (bound states), solutions exist only for... 

The presentation of a presumed exact bound state solution of the Schrodinger Coulomb Hamiltonian as a product of an electronic and a nuclear-motion part has been considered both by Hunter [9] and, more recently, by Gross [10]. Eor the present purposes, the Hunter approach will be employed on the translationally invariant form of the Hamiltonian, given earlier. Were the exact solution known. Hunter argues that it could be written in the form... 

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