Open shell molecules

IXDCf is faster than MINDO/3, MNDO, AMI, and PM3 and, unlike C XDO, can deal with spin effects. It is a particularly appealing choice for UHF calculations on open-shell molecules. It is also available for mixed mode calculations (see the previous section ). IXDO shares the speed and storage advantages of C XDO and is also more accurate. Although it is preferred for numerical results, it loses some of the simplicity and inierpretability of C XDO. 

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. 

Cl results can vary a little bit from one software program to another for open-shell molecules. This is because of the HF reference state being used. Some programs, such as Gaussian, use a UHF reference state. Other programs, such as MOLPRO and MOLCAS, use a ROHF reference state. The difference in results is generally fairly small and becomes smaller with higher-order calculations. In the limit of a full Cl, there is no difference. 

In most cases of closed-shell molecules Koopmans theorem is a reasonable approximation but N2 (see Section 8.1.3.2b) is a notable exception. For open-shell molecules, such as O2 and NO, the theorem does not apply. 

Berthier, G., Self-Consistent Field Methods for Open-Shell Molecules, in Molecular Orbitals in Chemistry, Physics, and Biology, P. O. Lbwdin and B. Pullman, Eds., Academic Press, New York, 1964, pp. 57-82. 

It is possible to construct a HF method for open-shell molecules that does maintain the proper spin symmetry. It is known as the restricted open-shell HF (ROHF) method. Rather than dividing the electrons into spin-up and spin-down classes, the ROHF method partitions the electrons into closed- and open-shell. In the easiest case of the high-spin wavefunction ( op = — electrons in op... 

Although the above discussion assumes that all MOs are occupied by two electrons, it turns out that the basic ideas can be extended to open-shell molecules in which there are unequal numbers of electrons in the two spin states. Without showing the complicated mathematics, we will show how the wavefunction can be determined by constructing two Fock matrices for each spin state and then solving two sets of coupled Roothaan equations ... 

Thomas Bally and Weston Thatcher Borden, Calculations on Open-Shell Molecules A Beginner s Guide. 

Fig. 3 A conceptual sketch showing approaches to molecular ferrimagnets (paired spins in the open-shell molecules have been left out).

Two theories were introduced in the 1960s on how to align electron spins in parallel between open-shell molecules (McConnell, 1963, 1967). 

The theory had never been tested on a logical model system. Let us consider in detail one representative case, the superimposable stacking of the two benzene rings, one from each triplet diphenylcarbene molecule. These are considered to represent idealized modes of dimeric interaction of the aromatic ring parts of open-shell molecules in ordered molecular assemblies like crystals, liquid crystals and membranes. 

The same Japanese crew has prepared open-shell molecules carrying the NN groups (Sakurai et al. 1996,2000 Kumai et al. 1996). These donors are presented in Scheme 1.34. The corresponding... 

Quantum chemical calculations have become a valuable tool in the research of reactive intermediates. Unfortunately, there is no unique computational method that can be uniformly applied in all cases and give an accurate answer at a practical cost. A variety of computational methods are available, each with its own weaknesses and advantages. The species that are of interest in this chapter and which often have unpaired electrons, pose specific problems in calculating their properties, and some care in choosing appropriate methods is necessary for obtaining meaningful results. In this respect, an excellent guide for calculations on open-shell molecules has been recently published [51]. 

Strength of positivity conditions Spin and spatial symmetry adaptation 1. Spin adaptation and S-representabiUty Open-shell molecules... 

After the energy is expressed as a functional of the 2-RDM, a systematic hierarchy of V-representabihty constraints, known as p-positivity conditions, is derived [17]. We develop the details of the 2-positivity, 3-positivity, and partial 3-positivity conditions [21, 27, 34, 33]. In Section II.E the formal solution of V-representability for the 2-RDM is presented through a convex set of two-particle reduced Hamiltonian matrices [7, 21]. It is shown that the positivity conditions correspond to certain classes of reduced Hamiltonian matrices, and consequently, they are exact for certain classes of Hamiltonian operators at any interaction strength. In Section II.F the size of the 2-RDM is reduced through the use of spin and spatial symmetries [32, 34], and in Section II.G the variational 2-RDM method is extended to open-shell molecules [35]. 

Also the variational 2-RDM method for computing the 2-RDM of a singlet state may be applied directly to open-shell molecules without significant modification. 

Electron affinities (EAs) are considerably more difficult to calculate than IPs. For example, EAs are much more sensitive to the basis set than the corresponding IPs. The EKT provides also the means for calculation of these magnitudes, but unfortunately the EKT-EA description is often very poor. On the other hand, vertical EAs can be calculated by the energy difference for neutral molecules (M°) and negative ions (M ) E(M°) — E M ) at near-experimental geometries of M°. Table IV lists the obtained vertical EAs for selected open-shell molecules. 

While the vast majority of molecules may be described in terms of closed-shell electron configurations, that is, all electrons being paired, there are several important classes of molecules with one or more unpaired electrons. So-called free radicals are certainly the most recognizable. One way to treat open-shell molecules is by strict analogy with the treatment of closed-shell molecules, that is, to insist that electrons are either paired or are unpaired. 

Unrestricted models, for example, the unrestricted Hartree-Fock (or UHF) model, are actually simpler and generally less costly than the corresponding restricted models, and because of this are much more widely used. Results for open-shell molecules provided in this book will make use of unrestricted models. 

For closed-shell molecules (in which all electrons are paired), the spin density is zero everywhere. For open-shell molecules (in which one or more electrons are unpaired), the spin density indicates the distribution of unpaired electrons. Spin density is an obvious indicator of reactivity of radicals (in which there is a single unpaired electron). Bonds will be made to centers for which the spin density is greatest. For example, the spin density isosurface for allyl radical suggests that reaction will occur on one of the terminal carbons and not on the central carbon. 

It is imperative to use CASSCF wave functions for singlet diradicals and other open-shell molecules for which a single configuration provides an inadequate description of the wave function. However, perhaps surprisingly, CASSCF calculations often perform rather poorly in calculations on molecules and TSs with closed shells of electrons, if the active electrons are delocalized. An example is... 

The main conclusion of this section is that the matrix elements of all terms in the collision Hamiltonian in the fully uncoupled space-fixed representation can be reduced to simple products of integrals of the type (8.46). Such matrix elements are very easy to evaluate numerically. The fiilly uncoupled representation is therefore very convenient for the development of the coupled channel codes for collision problems involving open-shell molecules with many angular momenta that need to be accounted for. The price for simplicity is a very large number of basis states that need to be included in the expansion of the eigenstates of the full Hamiltonian to achieve full basis set convergence (see Section 8.3.4). 

The total angular momentum basis is thus computationally more efficient, even for collision problems in external fields. There is a price to pay for this. The expressions for the matrix elements of the collision Hamiltonian for open-shell molecules in external fields become quite cumbersome in the total angular momentum basis. Consider, for example, the operator giving the interaction of an open-shell molecule in a 51 electronic state with an external magnetic field. In the uncoupled basis (8.43), the matrix of this operator is diagonal with the matrix elements equal to Mg, where is the projection of S on the magnetic field axis. In order to evaluate the matrix elements of this operator in the coupled basis, we must represent the operator 5 by spherical tensor of rank 1 (Sj = fl theorem [5]... 

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