Open shell systems

The Roothaan-Hall equations are not applicable to open-shell systems, which contain one or more unpaired electrons. Radicals are, by definition, open-shell systems as are some ground-state molecules such as NO and 02. Two approaches have been devised to treat open-shell systems. The first of these is spin-restricted Hartree-Fock (RHF) theory, which uses combinations of singly and doubly occupied molecular orbitals. The closed-shell approach that we have developed thus far is a special case of RHF theory. The doubly occupied orbitals use the same spatial functions for electrons of both a and spin. The orbital expansion Equation (2.144) is employed together with the variational method to derive the optimal values of the coefficients. The alternative approach is the spin-unrestricted Hartree-Fock (UHF) theory of Pople and Nesbet [Pople and Nesbet 1954], which uses two distinct sets of molecular orbitals one for electrons of a spin and the other for electrons of / spin. Two Fock matrices are involved, one for each type of spin, with elements as follows  

UHF theory also uses two density matrices, the full density matrix being the sum of these 

The summations in Equations (3.3) and (3 4) are over the occupied orbitals with a and (3 spin as appropriate. Thus, cnocc + Pocc equals the total number of electrons in the system. In a closed-shell Hartree-Fock wavefunction the distribution of electron spin is zero everywhere because the electrons are paired. In an open-shell system, however, there is an excess of electron spin, which can be expressed as the spin density, analogous to the electron density The spin density p (r) at a point r is given by  

The fact is that the molecular orbitals describing the resulting cation may well be quite different from those of the parent molecule. We speak of electron relaxation, and so we need to examine the problem of calculating accurate HF wavefunctions for open-shell systems. 

We would have P = 2R] and R2 = 0 for a closed-shell singlet state. The closed-shell electronic energy expression given earlier, 

I have introduced the occupation numbers vi and U2 (where ui = 2 and U2 = 1 in this simple case) to emphasize the symmetry of the electronic energy expression. 

For closed-shell states, we found an energy expression e = Tr(Ph,)+ iTr(PG) 

We then allow Ri and R2 to vary, subject to orthonormality, just as in the closed-shell case. Just as in the closed-shell case, Roothaan (1960) showed how to write a Hamiltonian matrix whose eigenvectors give the columns U] and U2 above. 

for the sake of argument, consider the case where there is a closed shell of ni doubly occupied orbitals, and an open shell of n2 orbitals, all of which are singly occupied with parallel spins. The LCAO-MOs of the closed shell and the open shell can be collected in the matrices Ui and U2, with n and n2 columns respectively, and we define density matrices Ri and R2 for each shell as 

Spin density is found in the molecular plane because of spin polarization, which is an effect arising from exchange correlation. The Fermi hole that surrounds the unpaired electron allows other electrons of the same spin to localize above and below the molecular plane slightly more than can electrons of opposite spin. Thus, if the unpaired electron is a, we would expect there to be a slight excess of fi density in the molecular plane as a result, the H hyperfine splitting should be negative (see Section 9.1.3), and this is indeed the situation observed experimentally. An ROHF wave function, because it requires the spatial distribution of both spins in the doubly occupied orbitals to be identical, cannot represent this physically realistic situation. 

To permit the a and ft spins to occupy different regions of space, it is necessary to treat them individually in the construction of the molecular orbitals. Following this formalism, we would rewrite our methyl radical wave function Eq. (6.6) as 

Finally, note that some open-shell systems cannot be described by a single determinant. The classical example is an open-shell singlet, i.e., a system having electrons of a and spin 

Besides being intuitively satisfying, ROHF theory produces wave functions that are eigenfunctions of the operator 5 (just as tire true wave function must be), having eigenvalues 5(5 -F 1) where 5 is the magnitude of the vector sum of the spin magnetic moments for all of the unpaired electrons. However, ROHF theory fails to account for spin polarization in 

MOs can be different, and this permits spin polarization. Equations (6.8) and (6.9) define umestricted Hartree-Fock (UHF) theory. 

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As fonnulated above, the FIF equations yield orbitals that do not guarantee that F has proper spin symmetry. To illustrate, consider an open-shell system such as the lithium atom. If Isa, IsP, and 2sa spin orbitals are chosen to appear in F, the Fock operator will be... 

The olassio papers in whioh the SCF equations for olosed- and open-shell systems are treated are ... 

WFth all semi-empirical methods, IlyperChem can also perform psendo-RIfF calculations for open -shell systems. For a doublet stale, all electrons except one are paired. The electron is formally divided into isvo "half electron s" with paired spins. Each halfelec-... 

Although LHF is often a better theoretical treatment of open-shell systems than the RHF (half-electron) methods, it takes longer to compute. Separate matrices for electrons of each spin roughly double the length of the calculation. ... 

Some systems converge poorly, particularly those with multiple bonds or weak interactions between open-shell systems. HyperChem includes two convergence accelerators. One is the default con verge rice accelerator, effective in speed in g up ri orm ally... 

Total spin den sity reflects th e excess probability of fin din g a versus P electrons in an open-shell system. Tor a system m which the a electron density is equal to the P electron density (for example, a closed-shell system), the spin density is zero. 

For sueh a funetion, the CI part of the energy minimization is absent (the elassie papers in whieh the SCF equations for elosed- and open-shell systems are treated are C. C. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951) 32, 179 (I960)) and the density matriees simplify greatly beeause only one spin-orbital oeeupaney is operative. In this ease, the orbital optimization eonditions reduee to ... 

There are a number of other technical details associated with HF and other ah initio methods that are discussed in other chapters. Basis sets and basis set superposition error are discussed in more detail in Chapters 10 and 28. For open-shell systems, additional issues exist spin polarization, symmetry breaking, and spin contamination. These are discussed in Chapter 27. Size-consistency and size-extensivity are discussed in Chapter 26. 

For an open-shell system, try converging the closed-shell ion of the same molecule and then use that as an initial guess for the open-shell calculation. Adding electrons may give more reasonable virtual orbitals, but as a general rule, cations are easier to converge than anions. 

Convergence problems are very common due to the number of orbitals available and low-energy excited states. The most difficult calculations are generally those with open-shell systems and an unfllled coordination sphere. All the techniques listed in Chapter 22 may be necessary to get such calculations to converge. 

Many transition metal systems are open-shell systems. Due to the presence of low-energy excited states, it is very common to experience problems with spin contamination of unrestricted wave functions. Quite often, spin projection and annihilation techniques are not sufficient to correct the large amount of spin contamination. Because of this, restricted open-shell calculations are more reliable than unrestricted calculations for metal system. Spin contamination is discussed in Chapter 27. 

This program is excellent for high-accuracy and sophisticated ah initio calculations. It is ideal for technically difficult problems, such as electronic excited states, open-shell systems, transition metals, and relativistic corrections. It is a good program if the user is willing to learn to use the more sophisticated ah initio techniques. 

LORG (localized orbital-local origin) technique for removing dependence on the coordinate system when computing NMR chemical shifts LSDA (local spin-density approximation) approximation used in more approximate DFT methods for open-shell systems LSER (linear solvent energy relationships) method for computing solvation energy... 

RHF (restricted Hartree-Fock) ah initio method for singlet systems ROHF (restricted open-shell Hartree-Fock) ah initio method for open-shell systems... 

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) ... 

It includes a significant number of molecules with unusual electronic states (for example, ions, open shell systems and hypervalent systems). 

So far, we have considered only the restricted Hartree-Fock method. For open shell systems, an unrestricted method, capable of treating unpaired electrons, is needed. For this case, the alpha and beta electrons are in different orbitals, resulting in two sets of molecular orbital expansion coefficients ... 

The two sets of coefficients result in two sets of Fock matrices (and their associated density matrices), and ultimately to a solution producing two sets of orbitals. These separate orbitals produce proper dissociation to separate atoms, correct delocalized orbitals for resonant systems, and other attributes characteristic of open shell systems. However, the eigenfunctions are not pure spin states, but contain some amount of spin contamination from higher states (for example, doublets are contaminated to some degree by functions corresponding to quartets and higher states). 

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. 

From the above it should be clear that UHF wave functions which are spin contaminated (more than a few percent deviation of (S ) from the theoretical value of S S + 1)) have disadvantages. For closed-shell systems an RHF procedure is therefore normally preferred. For open-shell systems, however, the UHF method has been heavily used. It is possible to use an ROHF type wave function for open-shell systems, but this leads to computational procedures which are somewhat more complicated than for the UHF case when electron correlation is introduced. 

P/h can be interpreted as an effective spin density of this open shell system. Similarly to the electron binding exjvession there is no first order contribution in the correlation potential, that is, = 0, so that 5 is correct through second order. However, the second order correction in the electron correction for... 

The contributions of the second order terms in for the splitting in ESR is usually neglected since they are very small, and in feet they correspond to the NMR lines detected in some ESR experiments (5). However, the analysis of the second order expressions is important since it allows for the calculation of the indirect nuclear spin-spin couplings in NMR spectroscoi. These spin-spin couplings are usually calcdated via a closed shell polarization propagator (138-140), so that, the approach described here would allow for the same calculations to be performed within the electron Hopagator theory for open shell systems. 

This article is an attempt to review possibilities in a quantum chemical treatment of open-shell systems. In order to cut down the extent of this review, we disregard some problems, especially those concerning macromolecules, polymerization reactions, and open-shell transition-metal complexes. Electron spin resonance is mentioned only briefly, because it has been a topic of many reviews. 

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