Schrodinger equation zero-order

In order to evaluate the above expression, solutions were found for the Schrodinger equation using the Morse potential for rotational quantum number i not equal to zero ... 

If the solution of the zero-order Schrodinger equation [i.e., all terms in (17) except V(r, Rq) are neglected] yields an/-fold degenerate electronic term, the degeneracy may be removed by the vibronic coupling terms. If T) and T ) are the two degenerate wave functions, then the vibronic coupling constant... 

The outlook given in this chapter on the theory of the second-order contracted Schrodinger equation and on its methodology has been aimed mostly at convincing the reader that this theory is not difficult to understand and that its methodology is now ready to be applied. That is, in the author s opinion, this methodology can be considered as accurate and probably more economical than the best standard quantum chemical computational methods for the study of states where the occupation number of spin orbitals is close to one or zero. 

We have split the Schrodinger equation into two independent second-order differential equations, one which depends only on the angles 6 and 0 and one which depends only on r. There are a number of solutions which satisfy our conditions 1 and 2 (continuity and good behavior for r — °°), but a condition for such solutions is that A = ( + 1), where is zero or an integer = 0,1,2,3,. 

In order to understand the wavefunction of an electron emitted from an atom by a certain ionization process, the wavefunction of a free particle with wavenumber k travelling along the positive z-axis will first be considered. The space and time dependence of this wavefunction follows from the time-dependent Schrodinger equation with zero potential1- and is given by... 

The Partitioning Technique.—Let P denote the projector onto some zero-order model wave function cP0> and Q its complement. The electronic Schrodinger equation... 

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 perfectly ordered crystals at absolute zero, solutions to the Schrodinger equation can be calculated on fast computers using density functional theory (DFT) based on the self-consistent local density approximation (LDA) simplifying procedures using different basis functions include augmented... 

Schrodinger equation have been performed using molecular orbital methods. The zero-order wave function is constructed as a single Slater determinant and the MOs are expanded in a set of atomic orbitals, the basis set. In a subsequent step the wave function may be improved by adding electron correlation with either Cl, MP or CC methods. There are two characteristics of such approaches (1) the one-electron functions, the -MQs, are delocalized over the whole molecule, and (2) an accurate treatment of the... 

Basis sets can be employed to solve derivative Schrodinger equations as naturally as employing them to solve the basic Schrodinger equations. An organized way of using basis sets, and a way that is quite suited for computational implementation, is to cast operators into their matrix representations in the given basis. This needs to be done for the zero-order Hamiltonian and for each derivative Hamiltonian operator. The zero-order Schrodinger equation for one state in matrix form is... 

Since this method has been described in the literature[16] we provide here only a brief outline of the numerical algorithm, but then discuss aspects of it in some detail. The application to H2O in Section HI should clarify the implementation and performance of the method. The algorithm has three parts definition of coordinates and coordinate groups and zero order Hamiltonians for each generation of an ESB from solutions of the zero order Hamiltonians and solution of the Schrodinger equation for many vibrational states efficiently by iterative methods. 

Using the normal product unperturbed Hamiltonian, the zero-order Schrodinger equation becomes... 

Equation (10.38) is recognized as the Schrodinger equation (4.13) for the one-dimensional harmonic oscillator. In order for equation (10.38) to have the same eigenfunctions and eigenvalues as equation (4.13), the function Slq) must have the same asymptotic behavior as in (4.13). As the intemuclear distance R approaches infinity, the relative distance variable q also approaches infinity and the functions F(R) and S(q) = RF(R) must approach zero in order for the nuclear wave functions to be well-behaved. As 7 —> 0, which is equivalent to q —Re, the potential U(q becomes infinitely large, so that F(R) and S(q rapidly approach zero. Thus, the function S(q) approaches zero as q -Re and as Roo. The harmonic-oscillator eigenfunctions V W decrease rapidly in value as x increases from x = 0 and approach zero as X —> oo. They have essentially vanished at the value of x corresponding to q = —Re. Consequently, the functions S(iq in equation (10.38) and V ( ) in... 

The electron-electron interaction is in the 1 IF method treated within the model of independent electrons. Within this approximation each electron moves in the average potential of other electrons. As a consequence, there is a non-zero probability that two electrons arc located at the same point in the space. Error resulting from this approximation is known as a correlation energy. The advantage of the model of independent electrons is that it allows to search for a wavefunction in the form of the product of one-electron functions (orbitals). Instead of a simple product function the Slater determinant is used in order to maintain anti-symmetry of the wavefunction. The solution of an n-electron Schrodinger equation can then be found in... 

Just as a satisfactory approximation to the exact solution of the time-independent Schrodinger equation for a many-electron molecule must be expressed in a form that belies the apparent simplicity of the spectrum, so too is the exact solution of the time-dependent Schrodinger equation often opaquely complex. For the time-independent problem, insight comes from defining a suitable zero-order model specifying the essential coupling terms (tT1)) and... 

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