Spin multiplicity, open-shell molecules

Recent advances in the techniques of photoelectron spectroscopy (7) are making it possible to observe ionization from incompletely filled shells of valence elctrons, such as the 3d shell in compounds of first-transition-series elements (2—4) and the 4/ shell in lanthanides (5, 6). It is certain that the study of such ionisations will give much information of interest to chemists. Unfortunately, however, the interpretation of spectra from open-shell molecules is more difficult than for closed-shell species, since, even in the simple one-electron approach to photoelectron spectra, each orbital shell may give rise to several states on ionisation (7). This phenomenon has been particularly studied in the ionisation of core electrons, where for example a molecule (or complex ion in the solid state) with initial spin Si can generate two distinct states, with spin S2=Si — or Si + on ionisation from a non-degenerate core level (8). The analogous effect in valence-shell ionisation was seen by Wertheim et al. in the 4/ band of lanthanide tri-fluorides, LnF3 (9). More recent spectra of lanthanide elements and compounds (6, 9), show a partial resolution of different orbital states, in addition to spin-multiplicity effects. Different orbital states have also been resolved in gas-phase photoelectron spectra of transition-metal sandwich compounds, such as bis-(rr-cyclo-pentadienyl) complexes (3, 4). 

The symmetric and antisymmetric squares have special prominence in molecular spectroscopy as they give information about some of the simplest open-shell electronic states. A closed-shell configuration has a totally symmetric space function, arising from multiplication of all occupied orbital symmetries, one per electron. The required antisymmetry of the space/spin wavefunction as a whole is satisfied by the exchange-antisymmetric spin function, which returns Fq as the term symbol. In open-shell molecules belonging to a group without... 

However, in addition to the need to run multiple SCF calculations to obtain the electronic energies, Ej, of many excited states, the ASCF approach has a number of undesirable features that severely limit its applicability. First, one encounters the problem of variational collapse. If a molecule possesses no symmetry elements, it is impossible to obtain SCF solutions for any state other than the lowest energy state of a given multiplicity. Second, many excited states cannot be adequately approximated by any single determinant and one must resort to low-spin restricted open-shell approaches which can be difficult to converge. 

In our first exploration of the T1 and T2 conditions [5] we obtained results of the RDM method for the ground-state energy and dipole moment for a collection of small molecules and molecular ions, both closed-shell and open-shell systems. (We don t mean closed shell in a strict sense, and we only constrained the spin and spin multiplicity eigenvalues, not the elements of the RDM.) The choice of molecules and configurations largely followed Ref. [18]—a paper that, we think, reinvigorated the classical RDM approach. We showed that the addition of the T1 and T2 conditions (T2 without the off-diagonal block X)... 

In section 7.4.2 we dealt with perhaps the simplest contribution to the effective Hamiltonian for a particular electronic state, that of the rotational kinetic energy. We now turn our attention to contributions which are only slightly more complicated, the so-called fine structure interactions involving the electron spin angular momentum S. Obviously for S to be non-zero, the molecule must be in an open shell electronic state with a general multiplicity (2S +1). 

Both OH and NO are open-shell free radicals, with doublet electron spin multiplicity. Consequently, the coupling of the angular momentum of the unpaired electron with the angular momentum N of nuclear rotation leads to a more complicated rotational energy level pattern than for a closed-shell molecule ( S electronic state) [45], For the upper, electronic state, the electron spin S = can couple with the rotational angular momentum to yield two fine-structure levels, with total angular momenta J= N + land N -1. These are... 

The electronic structures of molecules containing lanthanide and actinide atoms are extremely interesting due to the complex array of electronic states resulting from open f-shell electronic configurations. There are seven possible real projections for the 4f orbitals which could accommodate up to 14 electrons. As a result, numerous possible electronic states of varied spin multiplicities and spatial symmetries result even for a simple diatomic molecule consisting of a lanthanide or actinide element. A wealth of spectroscopic data has been accumulated up to now on several diatomics containing f-block elements. The spectra are considerably complex and definitive assignments are not always feasible. 

Treat states of different multiplicity, including open-shell states with many parallel spins, and positive and stable negative ions in addition to the neutral molecules. 

It is well-known that such functions can suffer from large amounts of spin contamination and are not suited to obtaining any surfaces except those that are the lowest of a given S3niraietry. However the UHF function, unlike an RHF function, will usually allow a molecule to separate correctly into its fragments for all decomposition channels. In contrast multi-reference-function techniques that include all configurations required to achieve correct separation would be intractable for even most three- and four-atom molecules. To limit the uncertainty introduced in using a UHF function for open shells, we monitor the multiplicity in the calculations. For some cases, such as the A A" state of HNO in the present paper, it offers a caution on the interpretation of the results, while for other cases, such as the A HCO surface, no multiplicity problems are encountered. 

The ground electronic states of H2, O2, and N2 are E states, while those of OH and NO are n states. These latter two molecules have open valence shells with net spin of 5, so that the multiplicity, 5, of these states is 2 (5 = 25 -i-1),... 

The mechanism of Zeeman relaxation in collisions of molecules in electronic states with nonzero electronic orbital angular momenta is different from that in collisions of E-state molecules. The response of non-E-state molecules to a magnetic field is determined by both the electron spin and the orbital angular momentum of the electrons in the open electronic shell. The orbital motion of the electrons induces electronic anisotropy, which gives rise to multiple adiabatic interaction potentials between the collision partners [20]. Consider, for example, the collision system of a hydrogen atom in an excited P state and a structureless atom, such as He. The interaction between the atoms can be described by an effective potential as a function of the interatomic distance and an angle between the direction of the electronic F-orbital and the interatomic separation line. The angular dependence of this potential is the electronic anisotropy. An alternative description of the interatomic interaction can be... 

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