Wavefunctions space

There is more than one solution to the SCF equations for the system, and the calculation procedure converges to a solution which is not the minimum (often a saddle point in wavefunction space). This indicates an RHF-to-RHF or UHF-to-UHF instability, depending on the wavefunction type. 

Recall that in his Theorems 3 and 4 Hans Kummer [3] defined a contraction operator, L, which maps a linear operator on A-space onto an operator on p-space and an expansion operator, E, which maps an operator on p-space onto an operator on A-space. Note that the contraction and expansion operators are super operators in the sense that they act not on spaces of wavefunctions but on linear spaces consisting of linear operators on wavefunction spaces. If the two-particle reduced Hamiltonian is defined as... 

Finally, one other non-linear wavefunction expansion will be described. If the geminals of the different electron pairs are further restricted to be identical for each electron pair, then the result is called the antisymmetrized product of identical geminals (APIG) wavefunctionThere are only (n—1) parameters in the APIG wavefunction which spans a subspace of the GP wavefunction space. Because of the severely restrictive form of this wavefunction, it has not been used extensively for MCSCF calculations but it has been used as a reference function for propagator calculationsfor which this wavefunction form has formal appeal. 

While semiempirical models which can be applied to molecules the size of 1 and 2 are necessarily only approximate, we were searching for trends rather than absolute values. In concept, the design of semiempirical quantum mechanical models of molecular electronic structure requires the definition of the electronic wavefunction space by a basis set of atomic orbitals representing the valence shells of the atoms which constitute the molecule. A specification of quantum mechanical operators in this function space is provided by means of parameterized matrices. Specification of the number of electrons in the system completes the information necessary for a calculation of electronic energies and wavefunctions if the molecular geometry is known. The selection of the appropriate functional forms for the parameterization of matrices is based on physical intuition and analogy to exact quantum mechanics. The numerical values of the parameters are obtained by fitting to selected experimental results, typically atomic properties. 

For systems containing two or more electrons of the same spin or other indistinguishable particles, an additional problem appears the node problem. For these systems, it is necessary to restrict the form of the total wavefunction (space and spin parts) such that it is antisymmetric to the exchange of electrons. For any electronic state other than the ground state, it is necessary to restrict further the properties of the wavefunction. The effect of these restrictions is the imposition of nodal surfaces, on which V /(X) = 0, in the space part of the wavefunction. The topic of nodal surfaces is discussed later in the section on Fixed-Node Calculations. 

To extract infomiation from the wavefimction about properties other than the probability density, additional postulates are needed. All of these rely upon the mathematical concepts of operators, eigenvalues and eigenfiinctions. An extensive discussion of these important elements of the fomialism of quantum mechanics is precluded by space limitations. For fiirther details, the reader is referred to the reading list supplied at the end of this chapter. In quantum mechanics, the classical notions of position, momentum, energy etc are replaced by mathematical operators that act upon the wavefunction to provide infomiation about the system. The third postulate relates to certain properties of these operators ... 

Nuclear pemuitations in the N-convention (which convention we always use for nuclear pemuitations) and rotation operations relative to a nuclear-fixed or molecule-fixed reference frame, are defined to transfomi wavefunctions according to (equation Al.4.56). These synnnetry operations involve a moving reference frame. Nuclear pemuitations in the S-convention, point group operations in the space-fixed axis convention (which is the convention that is always used for point group operations see section Al.4.2,2 and rotation operations relative to a space-fixed frame are defined to transfomi wavefiinctions according to (equation Al.4.57). These operations involve a fixed reference frame. 

The main drawback of the chister-m-chister methods is that the embedding operators are derived from a wavefunction that does not reflect the proper periodicity of the crystal a two-dimensionally infinite wavefiinction/density with a proper band structure would be preferable. Indeed, Rosch and co-workers pointed out recently a series of problems with such chister-m-chister embedding approaches. These include the lack of marked improvement of the results over finite clusters of the same size, problems with the orbital space partitioning such that charge conservation is violated, spurious mixing of virtual orbitals into the density matrix [170], the inlierent delocalized nature of metallic orbitals [171], etc. 

Even expression ( B3.4.31), altiiough numerically preferable, is not the end of the story as it does not fiilly account for the fact diat nearby classical trajectories (those with similar initial conditions) should be averaged over. One simple methodology for that averaging has been tln-ough the division of phase space into parts, each of which is covered by a set of Gaussians [159, 160]. This is done by recasting the initial wavefunction as... 

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, ... 

T indicates that the integration is over all space. Wavefunctions which satisfy this condition re said to be normalised. It is usual to require the solutions to the Schrodinger equation to be rthogonal ... 

VVc can now see why the normalisation factor of the Slater determinantal wavefunction is I v/N . If each determinant contains N terms then the product of two Slater determinants, ldeU rminant][determinant], contains (N ) terms. However, if the spin orbitals form an oi lhonormal set then oidy products of identical terms from the determinant will be nonzero when integrated over all space. We Ccm illustrate this with the three-electron example, k ljiiiidering just the first two terms in the expansion we obtain the following ... 

The fact that the set of hi is, in principle, complete in r-space allows the full (electronic and nuclear) wavefunction h to have its r-dependence expanded in terms of the hp... 

For all point, axial rotation, and full rotation group symmetries, this observation holds if the orbitals are equivalent, certain space-spin symmetry combinations will vanish due to antisymmetry if the orbitals are not equivalent, all space-spin symmetry combinations consistent with the content of the direct product analysis are possible. In either case, one must proceed through the construction of determinental wavefunctions as outlined above. 

All of the CSFs in the SCF (in which case only a single CSF is included) or MCSCF wavefunction that was used to generate the molecular orbitals (jii. This set of CSFs are referred to as spanning the reference space of the subsequent CI calculation, and the particular combination of these CSFs used in this orbital optimization (i.e., the SCF or MCSCF wavefunction) is called the reference function. 

A basis set is the mathematical description of the orbitals within a system (which in turn combine to approximate the total electronic wavefunction) used to perform the theoretical calculation. Larger basis sets more accurately approximate the orbitals by imposing fewer restrictions on the locations of the electrons in space. In the true quantum mechanical picture, electrons have a finite probability of existing anywhere in space this limit corresponds to the infinite basis set expansion in the chart we looked at previously. 

In cases where the potential is time-independent, we find that the wavefunction can be factorized into space- and time-dependent parts... 

Wavefunctions by themselves can be very beautiful objects, but they do not have any particular physical interpretation. Of more importance is the Bom interpretation of quantum mechanics, which relates the square of a wavefunction to the probability of finding a particle (in this case a particle of reduced mass /r vibrating about the centre of mass) in a certain differential region of space. This probability is given by the square of the wavefunction times dx and so we should concentrate on the square of the wavefunction rather than on the wavefunction itself. 

This simple wavefunction is antisymmetric to the exchange of electron names, and treats both space and spin. 

Some authors write x = r s to denote the total variables of the electron, and write the total wavefunction as k(x) or F(r, s). 1 have used a capital here to emphasize that the total wavefunction depends on both the space and spin variables. I will use the symbol dr to denote a differential space element, and ds to denote a differential spin element. 

In order to calculate the total probability (which comes to 1), we have to integrate over both space dr and spin ds. In the case of the hydrogen molecule-ion, we would write LCAO wavefunctions... 

Electronic wavefunctions symbolized in this text as I e(ri, S], ra, S2,..., r , s ) depend on the spatial (r) and spin (s) variables of all the m electrons. The electron density on the other hand depends only on the coordinates of a single electron. I discussed the electron density in Chapter 5, and showed how it was related to the wavefunction. The argument proceeds as follows. The chance of finding electron 1 in the differential space element dti and spin element ds] with the other electrons anywhere is given by... 

A vector space is a set with very special properties, which I don t have time to discuss. Wavefunctions are members of vector spaces. If we identify set A with the set of all possible electron densities for the problem of interest, and set B as the set of all real energies, then / defines a density functional. 

Because the square of any number is positive, we don t have to worry about i i having a negative sign in some regions of space (as a function such as sin x has) probability density is never negative. Wherever i , and hence i i2, is zero, the particle has zero probability density. A location where i]i passes through zero (not just reaching zero) is called a node of the wavefunction so we can say that a particle has zero probability density wherever the wavefunction has nodes. 

The expressions for a number of other atomic orbitals are shown in Table 1.2a (for R) and Table 1.2b (for Y). To understand these tables, we need to know that each wavefunction is labeled by three quantum numbers n is related to the size and energy of the orbital, / is related to its shape, and mt is related to its orientation in space. 

FIGURE U5 When two ls-orbitals overlap in the same region of space in such a way that their wavefunctions have the same signs in that region, their wavefunctions (red lines) interfere constructively and give rise to a region of enhanced amplitude between the two nuclei (blue line). 

Born interpretation The interpretation of the square of the wavefunction, i j, of a particle as the probability density for finding the particle in a region of space. 

---