Wave function linear combination

The wave function, obeying the Pauli principle, will be a linear combination of functions (9.6)... 

We start from a most general form [53-56] of the wave function. It differs from the GF approximation eq. (1.181) in that respect that the number of electrons in each group is not fixed, so that the generalized group function (GGF) expansion is a linear combination of functions which are antisymmetrized products of multipliers with a different number of electrons in the groups [53-56] ... 

The boundary conditions are periodic and the number of allowed values of the wave vector k is equal to the number of unit cells in the crystal. These eigenfunctions constitute the basis for the infinite-dimensional Hilbert space of the crystal Hamiltonian and any function with the same boundary conditions can be expressed as a linear combination of functions in this complete set. 

If Wn = Wk, so that fa and in are two linearly independent wave functions belonging to the same energy level, ik and in are not necessarily orthogonal, but it is always possible to construct two wave functions fa and fa belonging to this level which are mutually orthogonal. This can be done in an infinite number of ways by forming the combinations... 

The Hiickel MO method assumes that the a and n systems can be treated independently and then considers only the n system. The extended Hiickel theory (EHT), developed by Hoffmann, considers all of the valence electrons in the system, but it does not deal with the core electrons (those in orbitals below the valence shell). ° The EHT wave function is then written as a linear combination of functions describing each type of valence orbital on each atom in the structure. For hydrocarbons, the EHT wave fimctions include the 2s and all three 2p atomic orbitals of carbon as well as the Is orbitals of hydrogen atoms. The overlap integrals (S,y) are computed, whereas they were ignored in HMO theory. The elements of the secular determinant are assigned a value... 

Each HE molecular orbital is written as a linear combination of functions describing atomic orbitals. The entire set of such equations for the atomic orbitals in a molecule is called a basis set, and each equation is called a basis set function. Ideally, these basis set fimctions would have the properties of hydrogenic wave functions, particularly with regard to the radial dependence of electron density probability as a function of distance of the electron from the nucleus, r. A type of basis set function proposed by Slater uses a radial component incorporating and such functions are called Slater t)rpe orbitals (STOs). Gaussian type orbitals (GTOs) have radial dependence and are easier to solve analytically, but they do not describe the radial dependence of electron density as well as do the STOs. 

The method for improving the wave function by taking linear combinations of functions obtained from the various electron configurations belonging to the same symmetry type has been called configuration interaction (65, 71, 72). 

In this linear combination, the function has a distinguished meaning (co = 1) this is the approximation of the wave function of the considered system. Therefore, one can require that Yq does not have a contribution to the higher corrections and so on ... 

The two diabatic nuclear wave functions xf and x can be expressed as linear combinations of auxiliary nuclear wave functions and, respec-... 

MCSCF methods describe a wave function by the linear combination of M configuration state functions (CSFs), with Cl coefficients, Ck,... 

The electronic wave functions of the different spin-paired systems are not necessarily linearly independent. Writing out the VB wave function shows that one of them may be expressed as a linear combination of the other two. Nevertheless, each of them is obviously a separate chemical entity, that can he clearly distinguished from the other two. [This is readily checked by considering a hypothetical system containing four isotopic H atoms (H, D, T, and U). The anchors will be HD - - TU, HT - - DU, and HU -I- DT],... 

Adopting the view that any theory of aromaticity is also a theory of pericyclic reactions [19], we are now in a position to discuss pericyclic reactions in terms of phase change. Two reaction types are distinguished those that preserve the phase of the total electi onic wave-function - these are phase preserving reactions (p-type), and those in which the phase is inverted - these are phase inverting reactions (i-type). The fomier have an aromatic transition state, and the latter an antiaromatic one. The results of [28] may be applied to these systems. In distinction with the cyclic polyenes, the two basis wave functions need not be equivalent. The wave function of the reactants R) and the products P), respectively, can be used. The electronic wave function of the transition state may be represented by a linear combination of the electronic wave functions of the reactant and the product. Of the two possible combinations, the in-phase one [Eq. (11)] is phase preserving (p-type), while the out-of-phase one [Eq. (12)], is i-type (phase inverting), compare Eqs. (6) and (7). Normalization constants are assumed in both equations ... 

In this chapter, we resfiict the discussion to elementary chemical reactions, which we define as reactions having a single energy bamer in both dhections. As discussed in Section I, the wave function R) of any system undergoing an elementary reaction from a reactant A to a product B on the ground-state surface, is written as a linear combination of the wave functions of the reactant, A), and the product, B) [47,54] ... 

The task is now to calculate the structure and energy of the system in the transition state between A and B. Its wave function is assumed to be constmcted from a linear combination of the two. It is convenient to use VB terminology for this purpose. Let the wave function of A be denoted by a VB function A) and that of B by B). 

Now, consider a complex nucleai wave function given by a linear combination of the two real nuclear wave functions [42,53],... 

Note that only the polynomial factors have been given, since the exponential parts are identical for all wave functions. Of course, any linear combination of the wave functions in Eqs. (D.5)-(D.7) will still be an eigenfunction of the vibrational Hamiltonian, and hence a possible state. There are three such linearly independent combinations which assume special importance, namely,... 

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... 

HyperChem uses single detenu in am rather than spin-adapted wave fn n ction s to form a basis set for th e wave Fin ciion sin a con -figuration interaction expansion. That is, HyperChem expands a Cl wave function, m a linear combination of single Slater deterniinants P,... 

To this pom t, th e basic approxmi alien is th at th e total wave I lnic-tion IS a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ah miiio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ah mitio calculation. However, there are two main things to be considered m the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222). will require the few esl possible term s for an accurate representation of a molecular orbital. The second one is the speed of tW O-electron integral calculation. 

Because single-electron wave functions are approximate solutions to the Schroe-dinger equation, one would expect that a linear combination of them would be an approximate solution also. For more than a few basis functions, the number of possible lineal combinations can be very large. Fortunately, spin and the Pauli exclusion principle reduce this complexity. 

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