Restricted determinants

Seif-TeS I 6.13A Gasoline, which contains octane, may burn to carbon monoxide if the air supply is restricted. Determine the standard reaction enthalpy for the incomplete combustion of liquid octane in air to carbon monoxide gas and liquid water from the standard reaction enthalpies for the combustions of octane and carbon monoxide ... 

AFhe performance of various DFT methods based on a restricted determinant is shown in... 

In Subsection 2.3.3, we introduced a simple mnemonic device for writing down the energy of a single determinant, constructed from a set of spin orbitals xj, in terms of one-electron integrals over spin orbitals i fc i and antisymmetrized two-electron integrals over spin orbitals (/1 Here we will show how one can express, with equal ease, the energy of any restricted determinant, constructed from spin orbitals ja and in... 

We have described the spin of a single electron by the two spin functions a(this section we will discuss spin in more detail and consider the spin states of many-electron systems. We will describe restricted Slater determinants that are formed from spin orbitals whose spatial parts are restricted to be the same for a and p spins (i.e., xi = il iP ), Restricted determinants, except in special cases, are not eigenfunctions of the total electron spin operator. However, by taking appropriate linear combinations of such determinants we can form spin-adapted configurations which are proper eigenfunctions. Finally, we will describe unrestricted determinants, which are formed from spin orbitals that have different spatial parts for different spins (i.e., fjS ). 

Such spin orbitals are called restricted spin orbitals, and determinants formed from them are restricted determinants. In such a determinant a given spatial... 

We now consider open-shell restricted determinants. Open-shell determinants are not eigenfunctions of except when all the open-shell electrons have parallel spin, as in Fig. 2.12. As an illustration, let us consider the four singly excited determinants that arise in the minimal basis H2 model (see Eq. (2.76)). The open-shell determinants... 

With restricted spin orbitals and restricted determinants, the spatial orbitals are constrained to be identical for a and fi spins. For example, the restricted Hartree-Fock (RHF) ground state of the Li atom is... 

Unrestricted determinants are not eigenfunctions of 6. Moreover, they cannot be spin-adapted by combining a small number of unrestricted determinants as is the case for restricted determinants. Thus the UHF ground state (2.26S) of the Li atom is not a pure doublet as is the RHF ground state (2.264). Nevertheless, unrestricted wave functions are commonly used as a first approximation to doublet and triplet states. 

It is worthwhile to stress the merits of remote sensing performed with the aid of photographic emulsions. However, the restrictions are just as obvious, and not because only satellites that are returning to earth can use this technique. We can actually recover films from unmanned satellites - a costly procedure, but it has been done with types of satellites other than those in current use. The essential restriction determines the width of the spectral band within which photographic emulsions can operate. 

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