The molecular wavefunction

The Hamiltonian operator acts on to give the permitted energy levels for the molecule. But what is In the quantum-mechanical definition, is a function that contains all the information that is possible to know about a system. This information can be obtained by acting on with the appropriate operator, e.g. the Hamiltonian, to recover the total energy. 

A computationally convenient approach, but by no means the only one, is to approximate each atomic radial function by a sum of i x Gaussian functions, ip. These functions are then multiplied by an angular function in order to create the three-dimensional symmetry of the s, p, d, etc. spatial orbitals. This collection 

Clearly each chemical element, with its own unique set of orbitals, requires its own basis-set description, and in turn we will need to provide a basis set for every atom in our molecule. Each orbital is then given by a linear combination, using a so-called Slater determinant, which takes the general form  

the columns of the matrix describe the one-electron molecular orbitals (technically the spin orbitals the spatial orbitals described above are combined with an a or a (3 spin function), and the electrons are assigned to N rows. By using a matrix, we have taken care of the requirement highlighted in Section 3.2.1 that the wavefunction we construct must be antisymmetric. This follows since the value of the determinant shown in Eq. 3.5, when written out in full, includes positive and negative orbital contributions of the type shown in Eq. 3.3. Thus, should any electronic configuration attempt to include two electrons with the same spin in the same orbital, the determinant (and therefore the wavefunction) will vanish. 

A self-consistent loop is set up, with the weighting coefficients altered until the ground-state electronic configuration is obtained. Or, put another way, we need to minimize the change in energy with respect to the change in the basis-set coefficients, i.e. d /dc. 

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For Iran sition metals th c splittin g of th c d orbitals in a ligand field is most readily done using HHT. In all other sem i-ctn pirical meth -ods, the orbital energies depend on the electron occupation. HyperCh em s m oiccii lar orbital calcii latiori s give orbital cri ergy spacings that differ from simple crystal field theory prediction s. The total molecular wavcfunction is an antisymmetrized product of the occupied molecular orbitals. The virtual set of orbitals arc the residue of SCT calculations, in that they are deemed least suitable to describe the molecular wavefunction, ... 

As outlined in Section III.A, knowledge of the molecular wavefunction implies knowledge of the electron distribution. By setting a threshold value for this function, the molecular boundaries can be established, and the path is open to a definition of molecular shape. A quicker, but quite effective, approach to this entity is taken by assuming that each atom in a molecule contributes an electron sphere, and that the overall shape of a molecular object results from interpenetration of these spheres. The necessary radii can be obtained by working backwards from the results of MO calculations21, or from some kind of empirical fitting22. 

Like (5.15b), the multipole approximation (5.19) is dependent on the long-range assumption (5.14) both approximations fail (for different reasons) if the molecular wavefunctions f and fi" overlap appreciably. 

Knowledge of the molecular wavefunction enables us to determine the electron density at any given point in space. Here we inquire about the amount of electronic charge that can be associated in a meaningful way with each individual atom of a A -electron system. Our analysis covers Mulliken s celebrated population analysis [31], as well as a similar, closely related method. 

The Born-Oppenheimer adiabatic approximation is introduced by neglecting the coupling terms in eq. (4-6). The molecular wavefunctions then reduce to the simple product form... 

Our approach to the dynamics of complex electronic rearrangements, has been based on an eikonal representation of the molecular wavefunction. [40, 41, 42] In this representation, wavefunctions are written in the form x(q. Q, t)exp[iS(Q, t)/h], with a factorized exponential function of classical-like variables Q, where S is a classical-like mechanical action. It can be applied without detailed preliminary knowledge of electronic rearrangements,... 

The scientific interests of Huzinaga are numerous. He initially worked in the area of solid-state theory. Soon, however, he became interested in the electronic structure of molecules. He studied the one-center expansion of the molecular wavefunction, developed a formalism for the evaluation of atomic and molecular electron repulsion integrals, expanded Roothaan s self-consistent field theory for open-shell systems, and, building on his own work on the separability of many-electron systems, designed a valence electron method for computational studies on large molecules. 

The stationary states Fa) in Sections 2.1 and 2.2 represent general molecular states including all electronic (q) and all nuclear (Q) degrees of freedom. In this section we employ the Bom-Oppenheimer approximation in order to separate the molecular wavefunction into a nuclear part, ,n (Q), and an electronic part, Set(q Q), with the latter depending... 

In some cases, the HF wavefunction does not describe the molecular wavefunction well, even to a first approximation, and MP2 and CCSD(T) methods are also usually inappropriate. This happens especially often when two electrons are nominally paired but in practice are partially decoupled, as in a molecule with a covalent bond that is partly broken. This can happen if the bond is stretched (H2 at long bond lengths is a classic case where HF fails). 

Next we approximate the molecular wavefunction i/t as a linear combination of atomic orbitals (LCAO). The molecular orbital (MO) concept as a tool in interpreting electronic spectra was formalized by Mulliken23 starting in 1932 and building on earlier (1926) work by Hund24 [31] (recall that Mulliken coined the word... 

In the Bom interpretation (Section 4.2.6) the square of a one-electron wavefunction ij/ at any point X is the probability density (with units of volume-1) for the wavefunction at that point, and j/ 2dxdydz is the probability (a pure number) at any moment of finding the electron in an infinitesimal volume dxdydz around the point (the probability of finding the electron at a mathematical point is zero). For a multielectron wavefunction T the relationship between the wavefunction T and the electron density p is more complicated, being the number of electrons in the molecule times the sum over all their spins of the integral of the square of the molecular wavefunction integrated over the coordinates of all but one of the electrons (Section 5.5.4.5, AIM discussion). It can be shown [9] that p(x, y, z) is related to the component one-electron spatial wavefunctions ij/t (the molecular orbitals) of a single-determinant wavefunction T (recall from Section 5.2.3.1 that the Hartree-Fock T can be approximated as a Slater determinant of spin orbitals i/qoc and i// /i) by... 

In general, any satisfactory theoretical calculation of a nuclear coupling constant requires reliable calculation of the molecular wavefunction. As a consequence, a realistic approximation to the actual charge distribution in the carbohydrate molecule must presumably enter any theoretical model that attempts to provide a quantitative interpretation of solvent effects. The simplest treatments, and those that have been proposed most frequently to account for the solvent effect in the absence of specific effects, are those in which the solvent is treated as a continuum surrounding the solute molecule. Several different models where the solvent dependence of coupling interactions is related to the polarity of the medium have been proposed.78-79 The solvation theory80,81 has been successfully used within the FPT formalism to interpret the effect of solvent on Jc H and 3/CH. On the basis of this model, the Hamiltonian of a particular molecule includes the solvent-solute interaction term //so,v ... 

When the basis set contains many terms (effectively infinite), one obtains the best possible result from solution of the Hartree-Fock equation the Hartree-Fock limit. Improvements beyond this limit are most usually achieved by allowing the molecular wavefunction I7 to be a linear combination of antisymmetrized products of orbitals i,... 

It seems that atoms-in-molecules methods may develop in two separate directions. The basis of composite functions (112) is not the most suitable in which to expand the molecular wavefunction as many terms are needed in order to express adequately the distortion of the atoms upon the formation of molecules. Thus the OM calculations above require very large basis sets (18 composite functions for Li 2, 100 for HF, and 204 for LiF2) in order to obtain reliable results. The i.c.c. method, on the other hand, which does allow for some distortion, achieves in several instances as good or even better results with much smaller basis sets. An attempt has been made by Arai110 to develop a method of deformed atoms in molecules , in which the composite functions are multiplied through by a certain spatial function, 2(ri, r%. .., tn), to be determined, which expresses the desired distortion. Thus, in hydride molecules, the 1. wavefunction of H, which is of the form exp(— r), is multiplied by a function 2(r)=exp(—<5r), so that the combined function, exp[—(1 + <5)r], with Sx 0.2, is now an adequate representation of... 

This exact form of the molecular wavefunction [Pg.59]

Returning now to the problem of the calculation of nonradiative transition rates, it becomes obvious that much work remains to be done before these calculations can be performed in a routine manner to generate the rates of unmeasured processes. The accurate prediction of a nonradiative rate requires an accurate knowledge of the molecular wavefunctions (both electronic and vibrational), the density of states, and most important, the nuclear-coordinate dependence of the electronic wavefunctions. 

Moments and polarizabilities can also be obtained by the fixed-charge method [76]. This technique allows for the single-step incorporation of the nonuniform electric field contributions due to gradients and higher order field derivatives. One or more charges are placed around the molecule in regions where the molecular wavefunctions are negligible. It is important that the basis set used for the field-free molecule be the same as that used in the presence of the field and that the molecule basis be adequate to describe any... 

These results may be used to rationalize the experimental additivity schemes for the molecular tensors. Owing to the — factor of force and torque operators, the molecular wavefunction is essentially weighted in the environment of nucleus /, which could imply transferability of atomic terms from molecule to molecule in a series of structurally and chemically related homologues. 

Although the use of strokes to represent bonds between atoms in molecules comes from the nineteenth century, the electron pair concept as necessary for the understanding of chemical bonding was introduced by G.N. Lewis (1875-1946) in 1916 (ref. 90) following Bohr s, then recently proposed, model of the atom. Indeed, the Lewis model still lies at the basis of much of present-day chemical thinking, although it was advanced before both the development of quantum mechanics and the introduction of the concept of electron spin. In a more quantitative way, it found a natural theoretical extension in the valence-bond approximation to the molecular wavefunction, as expressed in terms of the overlap of (pure or hybridized) atomic orbitals to describe the pairing of electrons, coupled with the concept of electron spin. 

This process of constructing functions for the various resonant formulae, followed by an adequate combination of them, is mathematically more complex than the mathematics of molecular orbital theory. It is therefore understandable that, after the initial preference of chemists for the v.b. bond theory which has a closer relation to Lewis structures - especially due to the contribution of Linus Pauling - m.o. theory became increasingly popular. In addition, m.o. theory leads directly, not only to fundamental states (through the occupied m.o.), but also to excited states (through vacant m.o.) of molecules. In recent years, however, a new form of valence-bond theory has been developed that is more amenable to computation (spin-coupled valence-bond theory) in which the molecular wavefunction is expressed as a linear combination of all the coupling schemes of the various electrons corresponding to the same resultant spin (ref. 97). 

In this representation, the molecular wavefunction is expanded using the electronic wavefunctions with the contiguration fixed at the reference configuration Rq. This representation is called a crude adiabatic (CA) representation and the basis Ro) the electronic basis. The other representation, the Born-Oppenheimer (BO) representation, is defined as... 

The approximation with (36) and (37) is called a Born-Huang (BH) approximation. In this approximation, the molecular wavefunction is written as... 

Xa molecular orbital method (32). We used a code (33) where the molecular wavefunctions were expressed as a linear combination of atomic orbitals. The basis functions were obtained by numerically solving the Hartree-Fock-Slater equations in the atomic-like potentials derived from the spherical average of the molecular potential around the nuclei. Thus the atomic orbitals us as the basis functions were automatically optimized in the molecular potential (33). 

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