Unrestricted HF calculations

Another property calculated in the XH3—Y series is the X—Y bond dissociation energy. For this purpose the MP2 optimized geometries for the XH3 and Y (doublet spin) radicals were obtained using the unrestricted HF (UHF) method. For comparison to experiment, the electronic energy differences for the reaction... 

After the HFW integrals have been assembled, we then move on to the self-consistent field (SCF) procedure. For the most part this is the same as the HF version (10), with the exception of constructing the Fock matrix. The Fock matrix elements for an unrestricted HFW calculation are analogous to their HF counterparts and are given by... 

The most general version of Hartree-Fock (HF) theory, in which each electron is permitted to have its own spin and spatial wave function, is called unrestricted HF (UHF). Remarkably, when a UHF calculation is performed on most molecules which have an equal number of alpha and beta electrons, the spatial parts of the alpha and beta electrons are identical in pairs. Thus the picture that two electrons occupy the same MO with opposite spins comes naturally from this theory. A significant simplification in the solution of the Fock equations ensues if one imposes this natural outcome as a restriction. The form of HF theory where electrons are forced to occupied MOs in pairs is called restricted HF (RHF), and the resulting wave function is of the RHF type. A cal-... 

One way to deal with unpaired electrons is to use the unrestricted HF (UHF). Whereas regular ab initio calculations restrict the one-electron spatial orbitals to be identical for a- and (3-spin electrons (so-called restricted HF, RHF), in UHF the orbitals are allowed to be different in the SCF processes. Usually, the difference in the spatial orbitals for a and (3 electrons is only slight. Unfortunately, when applied to a radical, UHF stumbles in a pitfall (97). It is called spin contamination. Unrestricted HF wave functions cannot be trusted to correspond to pure spin states such as a doublet for radicals or a singlet or triplet for diradicals. Theoretically speaking, the UHF wave function may not be an eigenfunction of the spin operators. 

In this section, we briefly discuss some of the electronic structure methods which have been used in the calculations of the PE functions which are discussed in the following sections. There are variety of ab initio electronic structure methods which can be used for the calculation of the PE surface of the electronic ground state. Most widely used are Hartree-Fock (HF) based methods. In this approach, the electronic wavefunction of a closed-shell system is described by a determinant composed of restricted one-electron spin orbitals. The unrestricted HF (UHF) method can handle also open-shell electronic systems. The limitation of HF based methods is that they do not account for electron correlation effects. For the electronic ground state of closed-shell systems, electron correlation effects can be accounted for relatively easily by second-order Mpller-Plesset perturbation theory (MP2). In modern implementations of MP2, linear scaling with the size of the system has been achieved. It is thus possible to treat quite large molecules and clusters at this level of theory. 

Let us say a little more about these methods. The HF methods are divided into spin-restricted HF (RHF) and spin-unrestricted HF (UHF) methods. Closed-shell systems are almost always calculated using RHF. In this procedure, one set of molecular orbitals is calculated and pairs of electrons are entered to the lowest-energy orbitals. If the molecule has an odd number of electrons, one orbital will be singly occupied and the species is a radical (spin = 0.5, expectation value of the spin-squared operator =0.75). Inmost cases, however, radicals are calculated using the UHF formalism. UHF calculations determine two sets of molecular orbitals, one for each type of spin named alpha and beta. These MO sets are similar but not identical. A radical, for instance, has one more a than P electron. The UHF procedure is more flexible than RHF because the paired a and P orbitals, which correspond to doubly occupied MOs in the RHF formalism, need not be identical. So UHF allows for spin polarization but, on the other hand, spin-contamination occurs (i.e., states of higher spin are mixed into the wave function). 

Another study of systems containing one-electron a bonds, using unrestricted HF/6-31G and MP2/6-31G calculations, included the planar D2h and the perpendicular D2d B-B-bonded structures (a) and (b) in Fig. 2-58. One feature of the latter species is its high rotation barrier, which is caused by hyperconjugation in the perpendicular D2d form [3]. Structural parameters are given in Table 2/22. 

More physically correct than ROHF, and much easier to implement computationally, is another scheme called unrestricted Hartree-Fock (UHF), wherein the manifold of occupied MOs is not subdivided into closed and open shells. Instead, a standard HF program is used to carry out parallel sets of HF calculations on two different sets of MOs, one containing only the a and the other only the p electrons. The resulting pairs of UHF MOs for a and p electrons, which are identical in an ROHF calculation, have similar nodal properties, but they differ from each other in spatial detail. The restriction, Inherent in the ROHF scheme, that paired electrons of opposite spin occupy identical MOs is thus removed in UHF calculations. ... 

MP methods have been developed for both spin-restricted HF (RHF), spin-unrestricted HF (UHF), and restricted open-shell HF (ROHF) wavefunctions to investigate both closed- and open-shell systems (see Table 2). For the calculation of electron systems with multireference character such as biradicals various multireference state (MRS) MP methods have been developed (Table 2). All these methods describe atoms, molecules, and reaction systems in the gas phase. However, many chemical reactions take place in solution phases. For this purpose, MP methods are available that start from a solvent corrected wavefunction where mostly polarizable continuum models are used (see Self-consistent Reaction Field Methods). 

The energy-gradient method was first put into practice for closed-shell HF calculations, but it has recently been extended to spin-unrestricted and restricted-HF calculations on open-shell systems and to certain types of multiconfigurational SCF calculations. A method has been developed, based on the perturbed HF theory, to calculate the derivatives of MO coefficients. With this in hand, one is now able to calculate the energy gradient for a... 

Both HF and DFT calculations can be performed. Supported DFT functionals include LDA, gradient-corrected, and hybrid functionals. Spin-restricted, unrestricted, and restricted open-shell calculations can be performed. The basis functions used by Crystal are Bloch functions formed from GTO atomic basis functions. Both all-electron and core potential basis sets can be used. 

There are three main methods for calculating electron correlation Configuration Interaction (Cl), Many Body Perturbation Theory (MBPT) and Coupled Cluster (CC). A word of caution before we describe these methods in more details. The Slater determinants are composed of spin-MOs, but since the Hamilton operator is independent of spin, the spin dependence can be factored out. Furthermore, to facilitate notation, it is often assumed that the HF determinant is of the RHF type. Finally, many of the expressions below involve double summations over identical sets of functions. To ensure only the unique terms are included, one of the summation indices must be restricted. Alternatively, both indices can be allowed to run over all values, and the overcounting corrected by a factor of 1/2. Various combinations of these assumptions result in final expressions which differ by factors of 1 /2, 1/4 etc. from those given here. In the present book the MOs are always spin-MOs, and conversion of a restricted summation to an unrestricted is always noted explicitly. 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

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