Nonrelativistic Schrodinger equation

Quantum Mechanical Generalities.—It will be recalled that in nonrelativistic quantum mechanics the state of a particle at a given instant t is represented by a vector in Hilbert space (f)>. The evolution of the system in time is governed by the Schrodinger equation... 

Many chemical problems can be discussed by way of a knowledge of the electronic state of molecules. The electronic state of a molecular system becomes known if we solve the electronic Schrodinger equation, which can be separated from the time-independent, nonrelativistic Schrodinger equation for the whole molecule by the use of the Bom-Oppenheimer approximation D. In this approximation, the electrons are considered to move in the field of momentarily fixed nuclei. The nuclear configuration provides the parameters in the Schrodinger equation. 

The topological (or Berry) phase [9,11,78] has been discussed in previous sections. The physical picture for it is that when a periodic force, slowly (adiabatically) varying in time, is applied to the system then, upon a full periodic evolution, the phase of the wave function may have a part that is independent of the amplitude of the force. This part exists in addition to that part of the phase that depends on the amplitude of the force and that contributes to the usual, dynamic phase. We shall now discuss whether a relativistic electron can have a Berry phase when this is absent in the framework of the Schrodinger equation, and vice versa. (We restrict the present discussion to the nearly nonrelativistic limit, when particle velocities are much smaller than c.)... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nonrelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that are normally below the relativistic scale, the Berry phase obtained from the Schrodinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

The molecular time-independent nonrelativistic Schrodinger equation assumes the form... 

We would like to point out some steps of derivation of the nonrelativistic limit Hamiltonians by means of the Foldy-Wouthuyisen transformation (Bjorken and Drell, 1964). The method is based on the transformation of a relativistic equation of motion to the Schrodinger equation form. 

With the triples correction added, the error relative to experiment is still as large as 15 kJ/mol. More importantly, we are now above experiment and it is reasonable to assume that the inclusion of higher-order excitations (in particular quadruples) would increase this discrepancy even further, perhaps by a few kJ/mol (judging from the differences between the doubles and triples corrections). Extending the coupled-cluster expansion to infinite order, we would eventually reach the exact solution to the nonrelativistic clamped-nuclei electronic Schrodinger equation, with an error of a little more than 15 kJ/mol. Clearly, for agreement with experiment, we must also take into account the effects of nuclear motion and relativity. 

Up to this point, we have considered the nonrelativistic Schrodinger equation. However, to calculate AEs to an accuracy of a few kJ/mol, it is necessary to account for relativistic effects, even for molecules containing only hydrogen and first-row atoms. Fortunately, the major relativistic contributions to the AEs of such molecules - the mass-velocity (MV), one-electron Darwin (ID), and first-order spin-orbit (SO) terms - are easily obtained [58]. 

Nearly all kinetic isotope effects (KIE) have their origin in the difference of isotopic mass due to the explicit occurrence of nuclear mass in the Schrodinger equation. In the nonrelativistic Bom-Oppenheimer approximation, isotopic substitution affects only the nuclear part of the Hamiltonian and causes shifts in the rotational, vibrational, and translational eigenvalues and eigenfunctions. In general, reasonable predictions of the effects of these shifts on various kinetic processes can be made from fairly elementary considerations using simple dynamical models. 

The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics [1]. The simplest form is the Dirac-Coulomb (DC) Hamiltonian, where the nonrelativistic one-electron terms in the Schrodinger equation are replaced by the one-electron Dirac operator hj). 

One simple form of the Schrodinger equation—more precisely, the time-independent, nonrelativistic Schrodinger equation—you may be familiar with is Hx i = ty. This equation is in a nice form for putting on a T-shirt or a coffee mug, but to understand it better we need to define the quantities that appear in it. In this equation, H is the Hamiltonian operator and v i is a set of solutions, or eigenstates, of the Hamiltonian. Each of these solutions,... 

The kinetic energy operator in the Schrodinger equation corresponds to the quadratic term in this nonrelativistic expansion, and thus the Schrodinger equation describes only the leading nonrelativistic approximation to the hydrogen energy levels. 

On the nonrelativistic quantum level, both the time-independent and time-dependent Schrodinger equations can be used to demonstrate the existence of RFR. As shown by Sakurai [68], the time-independent Schrodinger-Pauli equation can be used to demonstrate ordinary ESR and NMR in the nonrelativistic quantum limit. This method is adopted here to demonstrate RFR in nonrelativistic quantum mechanics with the time-independent Schrodinger-Pauli equation [68] ... 

We have therefore derived a nonrelativistic Schrodinger equation for a free particle with an additional negative potential energy term V = —jmc2. In order to apply this method to the hydrogen atom, the relevant Schrodinger... 

This article deals with QBSs in nonrelativistic quantum mechanics. Suppose that a QBS satisfies the time-independent Schrodinger equation with a Hamiltonian H. One may extract a normalizable wavefunction out of T such that in the large-amplitude region, by writing... 

The Schrodinger equation that we have been talking about is actually the time-independent (and nonrelativistic) Schrodinger equation the variables in the equation are spatial coordinates, or spatial and spin coordinates (Section 5.2.3.1) when... 

These formulas are valid for the nonrelativistic one-electron Schrodinger equation. The Lagrangian density is... 

All "conservative holonomic" systems must satisfy (at nonrelativistic speeds) the time-dependent Schrodinger equation ... 

In the nonrelativistic Schrodinger equation, time is a parameter, not a coordinate. Therefore the typical uncertainty relation relating the lifetime At and the half-width AE of the transition ... 

For the computational investigation of molecular systems containing heavy atoms, such as transition metals, lanthanides, and actinides, we could neglect neither relativity nor electron correlation. Relativistic effects, both spin-free and spin-orbit, increase with the nuclear charge of atoms. Therefore, instead of the nonrelativistic Schrodinger equation, we must start with the Dirac equation, which has four-component solutions. For many-electron systems, the four-component Hamiltonian is constructed from the one-electron Dirac operator with an approximated relativistic two-electron operator, such as the Coulomb, Breit, or Gaunt operator, within the nopair approximation. The four-component method is relativistically rigorous, which includes both spin-free and spin-orbit effects in a balanced way. However it requires much computational time since it contains more variational parameters than the approximated, one or two-component method. 

Relativistic effects may be also considered by other methods than pseudopotentials. It is possible to carry out relativistic all-electron quantum chemical calculations of molecules. This is achieved by various approximations to the Dirac equation, which is the relativistic analogue to the nonrelativistic Schrodinger equation. We do not want to discuss the mathematical details of this rather complicated topic, which is an area where much progress has been made in recent years and where the development of new methods is a field of active research. Interested readers may consult published reviews . A method which has gained some popularity in recent years is the so-called Zero-Order Regular Approximation (ZORA) which gives rather accurate results ". It is probably fair to say that... 

For a heavy element whose atomic number is beyond 50, the relativistic effects (error caused by the nonrelativistic approximation) on the valence state can not be ignored. In such a case, it is necessary to solve Dirac equation instead of nonrelativistic Schrodinger equation usually used for the electronic state calculation. The relativistic effects... 

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