Wave functions definition

From these rotation relationships and wave function definition are deduced the two models of the wave function, the differential one and the integral one resulting from the definition of the exponential function seen in Section 10.5.4 in Chapter 10 ... 

In an Abelian theory [for which I (r, R) in Eq. (90) is a scalar rather than a vector function, Al=l], the introduction of a gauge field g(R) means premultiplication of the wave function x(R) by exp(igR), where g(R) is a scalar. This allows the definition of a gauge -vector potential, in natural units... 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

By using the determinant fomi of the electronic wave functions, it is readily shown that a phase-inverting reaction is one in which an even number of election pairs are exchanged, while in a phase-preserving reaction, an odd number of electron pairs are exchanged. This holds for Htickel-type reactions, and is demonstrated in Appendix A. For a definition of Hilckel and Mbbius-type reactions, see Section III. 

These new wave functions are eigenfunctions of the z component of the angular momentum iij = —with eigenvalues = +2,0, —2 in units of h. Thus, Eqs. (D.l 1)-(D.13) represent states in which the vibrational angular momentum of the nuclei about the molecular axis has a definite value. When beating the vibrations as harmonic, there is no reason to prefer them to any other linear combinations that can be obtained from the original basis functions in... 

The original definition of natural orbitals was in terms of the density matrix from a full Cl wave function, i.e. the best possible for a given basis set. In that case the natural orbitals have the significance that they provide the fastest convergence. In order to obtain the lowest energy for a Cl expansion using only a limited set of orbitals, the natural orbitals with the largest occupation numbers should be used. 

The seeond term disappears since the Cl wave function is variational in the state coefficients, eq. (10.33). The three terms involving the derivative of the MO coefficients (dc/dX) also disappear owing to our choice of the Lagrange multipliers, eq. (10.36). If we furthermore adapt the definition that dH/dX = Pi (eq. (10.17)), the final derivative may be written as... 

Since aj always is positive or zero, and Ei — Eq always is positive or zero ( o is by definition the lowest energy), this completes the proof. Furthermore, in order for the equal sign to hold, all =0 since , o - q is non-zero (neglecting degenerate ground states). This in turns means that ao = ], owing to the nonnalization of, and consequently the wave function is the exact solution. 

The notion of electrons in orbitals consists essentially of ascribing four distinct quantum numbers to each electron in a many-electron atom. It can be shown that this notion is strictly inconsistent with quantum mechanics (7). Definite quantum numbers for individual electrons do not have any meaning in the framework of quantum mechanics. The erroneous view stems from the original formulation of the Pauli principle in 1925, which stated that no two electrons could share the same four quantum numbers (8), This version of the principle was superseded by a new formulation that avoids any reference to individual quantum numbers for separate electrons. The new version due to the independent work of Heisenberg and Dirac in 1926 states that the wave function of a many-electron atom must be antisymmetrical with respect to the interchange of any two particles (9,10). 

The definitions are here given under the assumption that the wave function XP is either antisymmetric or symmetric for a trial function without symmetry property, one has to replace the binomial factor NCV before the integrand by a factor l/p and sum over the N(N—l). . . (N—p+l) possible integrals which are obtained by placing the fixed coordinates x, x 2,. . ., x P in various ways in the N places of the first factor W and the fixed coordinates xv x2,. . xv similarly in the second factor W. By using Eq. II.8 we then obtain... 

In conclusion, we observe that many writers in the modern literature seem to agree about the convenience of the definition (Eq. 11.67), but that there has also been a great deal of confusion. For comparison we would like to refer to Slater, and Arai (1957). Almost the only exception seems to be Green et al. (1953, 1954), where the exact wave function is expanded as a superposition of orthogonal contributions with the HF determinant as its first term ... 

It is apparent that the Hartree-Fock level is characterized by an enormous average deviation from experiment, but standard post-HF methods for including correlation effects such as MP2 and QCISD also err to an extent that renders their results completely useless for this kind of thermochemistry. We should not, however, be overly disturbed by these errors since the use of small basis sets such as 6-31G(d) is a definite no-no for correlated wave function based quantum chemical methods if problems like atomization energies are to be addressed. It suffices to point out the general trend that these methods systematically underestimate the atomization energies due to an incomplete recovery of correlation effects, a... 

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