Orbital spatial

To obtain the force constant for constructing the equation of motion of the nuclear motion in the second-order perturbation, we need to know about the excited states, too. With the minimal basis set, the only excited-state spatial orbital for one electron is... 

By expanding the spatial orbitals into atomic orbitals and manipulating them properly, we have... 

A restricted Hartrec-Fock description means that spin-up and spin -down electron socciipy the same spatial orbitals ip,—there is no allowance for different spatial orbitals for different electron spins. 

I h e n otation h ere rn ean s th at electron 1 occupies a spatial orbital tpi with spin up (no bar on top ] electron 2 occupies spatial orbital tPi with spin down (a baron top), and so on. An RIIF description of the doublet S=l/2 slate obiamed by adding an electron to would be... 

At normal bond lengths, the LHK solution usually degenerates to the situation where the two spatial orbitals become identical. The LHK solution for Hp, for example, has a smooth potential energy... 

X spin orbital (product of spatial orbital and a spin function)... 

Another way of constructing wave functions for open-shell molecules is the restricted open shell Hartree-Fock method (ROHF). In this method, the paired electrons share the same spatial orbital thus, there is no spin contamination. The ROHF technique is more difficult to implement than UHF and may require slightly more CPU time to execute. ROHF is primarily used for cases where spin contamination is large using UHF. 

Open shell systems—for example, those with unequal numbers of spin up and spin down electrons—are usually modeled by a spin unrestricted model (which is the default for these systems in Gaussian). Restricted, closed shell calculations force each electron pair into a single spatial orbital, while open shell calculations use separate spatial orbitals for the spin up and spin down electrons (a and P respectively) ... 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

SCVB wave functions to include electron correlation is due to the fact that the VB orbitals are strongly localized, and since they are occupied by only one electron, they have the built-in feature of electrons avoiding each other. In a sense, an SCVB wave function is tte best wave function that can be constructed in terms of products of spatial orbitals. By allowing the orbitals to become non-orthogonal, the large majority (80-90%) of what is called electron correlation in an MO approach can be included in a single determinant wave function composed of spatial orbitals, multiplied by proper spin cou ing functions. 

Again the left superscript indicates the spin-triplet nature of the arrangement. The letter A means that it is spatially (orbitally) one-fold degenerate and it is upper-case because we describe two-electron wavefunctions. The subscript is g because the product of d orbitals is even under the octahedral centre of inversion, and the right subscript 2 must remain a mystery for us once again. 

It can easily be shown that the HF approximation discussed in Chapter 1 does include the Fermi-correlation, but completely neglects the Coulomb part. To demonstrate this, we analyze the Hartree-Fock pair density for a two-electron system with the two spatial orbitals ()> and < )2 and spin functions a and o2... 

In the former, electrons are assigned to orbitals in pairs, the total spin is zero, so the multiplicity is 1. In this case, the restricted Hartree-Fock method (RHF) can be applied. For radicals with doublet or triplet states, the unrestricted Hartree-Fock (UHF) has to be applied. In this method, a and, 3 electrons (spin up and spin down) are assigned to different spatial orbitals, so there are two distinct sets I and FJf... 

In these expressions, n,m> is the number (0,1, or 2) of electrons occupying spatial orbital 0,-(O) in F0), and is a variational trial function (orthogonal to 0,-(O)) for the first-order orbital correction The expressions (1.20) allow us to treat the perturbative effects on an orbital-by-orbital basis, isolating the corrections associated with each HF orbital ,. Equations (1.18)-(1.20) involve only s/ng/e-electron operators and integrations, and are therefore considerably simpler than (1.5c) and (1.5d). 

In Eq. (1.16a), A is the antisymmetrizer operator that converts the orbital product into a Slater determinant, insuring satisfaction of the Pauli exclusion principle. In this equation (alone), the same spatial orbital might appear twice, with different indices to indicate the change in spin. For example, / i (0,(7 ypf HA) might be the same as i<0)(F K/>,0,0" 2). a doubly occupied spatial orbital (n]m> = 2), with a bar denoting opposite spin in the second spin-orbital. 

It is possible to divide electron correlation as dynamic and nondynamic correlations. Dynamic correlation is associated with instant correlation between electrons occupying the same spatial orbitals and the nondynamic correlation is associated with the electrons avoiding each other by occupying different spatial orbitals. Thus, the ground state electronic wave function cannot be described with a single Slater determinant (Figure 3.3) and multiconfiguration self-consistent field (MCSCF) procedures are necessary to include dynamic electron correlation. 

The conclusion above that optimisation of the non-linear parameters in the AO basis leads to a basis with correct spatial symmetry properties cannot be true for all intemuclear separations. At R = 0 the orbital basis must pass over into the double-zeta basis for helium i.e. two different 1 s orbital exponents. It would be astonishing if this transition were discontinuous at R = 0. While considering the variation of basis with intemuclear distance it is worth remembering that the closed-shell spin-eigenfunction MO method does not describe the molecule at all well for large values of R the spin-eigenfunction constraint of two electrons per spatial orbital is completely unrealistic at large intemuclear separation. With these facts in mind we have therefore computed the optimum orbital exponents as a function of R for three wave functions ... 

Sebastian has emphasized that (17a) implies Pi <0.5 (since 0restricted form the wavefunction (11) has when the a and / -spin orbitals are constrained to be equal. It can be circumvented by removing this constraint and using different spatial orbitals for electrons with different spin, which is accomplished by making different choices for the coupling functions. 

Rather loosely, this may be thought of as taking U ao, so that double occupancy of 0) is totally forbidden. Because the wavefunction (11) has different a and /9-spin spatial orbitals, it is not a pure spin state (it will be a mixture of singlet, triplet, quintet, etc.). However, for the problem in hand, this does not seem to be much of a disadvantage. 

The 2-RDM/or the radical may be computed from the N + l)-electron density matrix for the dissociated molecule by integrating over the spatial orbital and spin associated with the hydrogen atom and then integrating over N — 2 electrons. Because the radical in the dissociated molecule can exist in a doublet state with its unpaired electron either up or down, that is, M = the 2-RDM for the radical is an arbitrary convex combination... 

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