Variational principles Schrodinger

You will see shortly that an exact solution of the electronic Schrodinger equation is impossible, because of the electron-electron repulsion term g(ri, r2). What we have to do is investigate approximate solutions based on chemical intuition, and then refine these models, typically using the variation principle, until we attain the required accuracy. This means in particular that any approximate solution will not satisfy the electronic Schrodinger equation, and we will not be able to calculate the energy from an eigenvalue equation. First of all, let s see why the problem is so difficult. 

The eigenvalues E0, Elt E2,. .. of the Schrodinger equation (Eq. II. 1) form the electronic energies of the system under consideration. It is evident that the solution of Eq. II. 1 must involve considerable mathematical difficulties, and so far, the strongest tool we know for handling this problem is the variation principle. If the wave function W is properly normalized so that... 

A key to the application of DFT in handling the interacting electron gas was given by Kohn and Sham [51] who used the variational principle implied by the minimal properties of the energy functional to derive effective singleparticle Schrodinger equations. The functional F[ ] can be split into four parts ... 

Obviously, the BO or the adiabatic states only serve as a basis, albeit a useful basis if they are determined accurately, for such evolving states, and one may ask whether another, less costly, basis could be just as useful. The electron nuclear dynamics (END) theory [1-4] treats the simultaneous dynamics of electrons and nuclei and may be characterized as a time-dependent, fully nonadiabatic approach to direct dynamics. The END equations that approximate the time-dependent Schrodinger equation are derived by employing the time-dependent variational principle (TDVP). 

The Variation Principle is the main point of departure all questions of symmetry, approximation etc. are judged from the point of view of their likely effect on the variational form of the Schrodinger equation. We attempt to take the minimal basis AO expansion method as far as possible while remaining within a family of well-defined conceptual models of the electronic structure which is theoretically and numerically underpinned by the variation principle. 

Eq. (22) have been derived from the variation principle alone (given the structure of H) they contain only the single model approximation of Eq. (9) the typically chemical idea that the electronic structure of a complex many-electron system can be (quantitatively as well as qualitatively) understood in terms of the interactions among conceptually identifiable separate electron groups. In the discussion of the exact solutions of the Schrodinger equation for simple systems the operators which commute with the relevant H ( symmetries ) play a central role. We therefore devote the next section to an examination of the effect of symmetry constraints on the solutions of (22). 

We therefore conclude that attempts to impose constraints which are based on the formal properties of the exact Schrodinger equation leads to contradictory and even self-contradictory results besides placing unnecessary limitations on the action of the variation principle. That is not to advocate wilful inconsistency, of course. We shall insist on consistencies but within the confines of our orbital-basis variational model. 

Variational Principles for the Time-Independent Wave-Packet-Schrodinger and Wave-Packet-Lippmann-Schwinger Equations. 

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]. 

But there is a more basic difficulty in the Hohenberg-Kohn formulation [19-21], which has to do with the fact that the functional iV-representability condition on the energy is not properly incorporated. This condition arises when the many-body problem is presented in terms of the reduced second-order density matrix in that case it takes the form of the JV-representability problem for the reduced 2-matrix [19, 22-24] (a problem that has not yet been solved). When this condition is not met, an energy functional is not in one-to-one correspondence with either the Schrodinger equation or its equivalent variational principle therefore, it can lead to energy values lower than the experimental ones. 

Now, using the variational principle with eq.(4) it can be shown that the electronic wave function H(p) fulfills the Schrodinger equation... 

Approximate solutions of the time-dependent Schrodinger equation can be obtained by using Frenkel variational principle within the PCM theoretical framework [17]. The restriction to a one-determinant wavefunction with orbital expansion over a finite atomic basis set leads to the following time-dependent Hartree-Fock or Kohn-Sham equation ... 

Defining the functional V = (i/ v f). where u(r) is a local potential for which E = T + V is bounded below, the Schrodinger variational principle requires E to be stationary subject to normalization =. The variation SE induced... 

The Schrodinger variational principle requires (T //1T) to be stationary subject to constant normalization ( ). Introducing a Lagrange multiplier, the variational condition is... 

In bound-state calculations, the Rayleigh-Ritz or Schrodinger variational principle provides both an upper bound to an exact energy and a stationary property that determines free parameters in the wave function. In scattering theory, the energy is specified in advance. Variational principles are used to determine the wave function but do not generally provide variational bounds. A variational functional is made stationary by choice of variational parameters, but the sign of the residual error is not determined. Because there is no well-defined bounded quantity, there is no simple absolute standard of comparison between different variational trial functions. The present discussion will develop a stationary estimate of the multichannel A -matrix. Because this matrix is real and symmetric for open channels, it provides the most... 

The theory of the / -matrix can be understood most clearly in a variational formulation. The essential derivation for a single channel was given by Kohn [202], as a variational principle for the radial logarithmic derivative. If h is the radial Hamiltonian operator, the Schrodinger variational functional is... 

The derivation by Kohn [202] can readily be extended to the multichannel 7 -matrix. The underlying logic depends on the variational principle that the multichannel Schrodinger functional is stationary for variations of a trial function that satisfies the correct boundary conditions if and only if that function satisfies the Schrodinger equation. In a matrix notation, suppressing summations and indices, the variational functional of Schrodinger is... 

Variational methods [6] for the solution of either the Schrodinger equation or its perturbation expansion can be used to obtain approximate eigenvalues and eigenfunctions of this Hamiltonian. The Ritz variational principle,... 

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