Many-electron basis functions

If the solution 4>(t) to the time-dependent Schrodinger equation is expanded as a linear combination of time-independent orthonormal many-electron basis functions i.e. 

We can try to build up solutions to (51) from a set of many-electron basis functions which satisfy the equations ... 

Consider two many-electron basis functions i and , interacting through the Hamiltonian A ... 

A straightforward generalisation of the above perturbation Cl/SO treatment is to use the correlated scalar relativistic eigenfimctions 4> ) of the scalar Hamiltonian /f as a truncated set of contracted many-electron basis functions for the total Hamiltonian. Introducing the subscript im for a given wave functions to mark out the spatial and spin degenerate components of this multi-plet, the matrix representation of the Hamiltonian writes... 

Multiple PESs may be of simultaneous interest based not only on physical reasons, as emphasized in the previous paragraph, but also for mathematical or computational reasons. Consider the basic paradigm of state-selective methods the nondynamical electron correlation for a specific state is calculated within a model space and then the dynamical electron correlation is calculated. The implicit assumption is that the zero-order model space many-electron basis functions (MEBFs) (e.g., MCSCF functions and MCSCF complementary space... 

As mentioned in Chapter 5, one can think of the expansion of an unknown MO in terms of basis functions as describing the MO function in the coordinate system of the basis functions. The multi-determinant wave function (4.1) can similarly be considered as describing the total wave function in a coordinate system of Slater determinants. The basis set determines the size of the one-electron basis (and thus limits the description of the one-electron functions, the MOs), while the number of determinants included determines the size of the many-electron basis (and thus limits the description of electron correlation). 

In developing perturbation theory it was assumed that the solutions to the unpermrbed problem formed a complete set. This is general means that there must be an infinite number of functions, which is impossible in actual calculations. The lowest energy solution to the unperturbed problem is the HF wave function, additional higher energy solutions are excited Slater determinants, analogously to the Cl method. When a finite basis set is employed it is only possible to generate a finite number of excited determinants. The expansion of the many-electron wave function is therefore truncated. 

Individual molecular orbitals, which in symmetric systems may be expressed as symmetry-adapted combinations of atomic orbital basis functions, may be assigned to individual irreps. The many-electron wave function is an antisymmetrized product of these orbitals, and thus the assignment of the wave function to an irrep requires us to have defined mathematics for taking the product between two irreps, e.g., a 0 a" in the Q point group. These product relationships may be determined from so-called character tables found in standard textbooks on group theory. Tables B.l through B.5 list the product rules for the simple point groups G, C, C2, C2/, and C2 , respectively. 

These nine basis functions r/j (5 on C and 4 x 1 = 4 on H) create nine spatial MO s ij/, which could hold 18 electrons for the ten electrons of CH4 we need only five spatial MO s. There is no upper limit to the size of a basis set there are commonly many more basis functions, and hence MO s, than are needed to hold all the electrons, so that there are usually many unoccupied MO s. In other words, the number of basis functions m in the expansions (5.52) can be much bigger than the... 

The many-electron wave function in a crystal forms a basis for some irreducible representation of the space group. This means that the wave function, with a wave vector k, is left invariant under the symmetry elements of the crystal class (e.g. translations, rotations, reflections) or transformed into a new wave function with the same wave vector k. 

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