Heavy-atom molecules

Pitzer, R.M. and Winter, N.W (1991) Electronic-Structure Methods for Heavy-Atom Molecules. International Journal of Quantum Chemistry, 40, 773-780. 

A bond separation reaction takes any molecule comprising three or more heavy (non-hydrogen) atoms into the set of simplest (two-heavy-atom) molecules containing the same bonds. The only requirement is that bonding must be defined in terms of a single valence (Lewis) structure or set of equivalent valence structures. This in turn guarantees that the bond separation energy is unique. ... 

As previously described in Chapter 6, a bond separation reaction breaks down any molecule comprising three or more heavy (nonhydrogen) atoms, and which can be represented in terms of a classical valence structure, into the simplest set of two-heavy-atom molecules containing the same component bonds. For example, the bond separation reaction for methylhydrazine breaks the molecule into methylamine and hydrazine, the simplest molecules incorporating CN and NN single bonds, respectively. 

There is no obvious best solution to these problems. One direction is to redefine (or generalize) the bond separation reaction such that the products are not restricted to the smallest (two-heavy-atom) molecules, but rather include molecules made up of larger components as well (functional groups, rings, etc.). For example, were the phenyl ring and the carboxylic acid functional group included as fragments , then the bond separation reaction for w-toluic acid could be written. 

Bond Separation Reaction. An Isodesmic Reaction in which a molecule described in terms of a conventional valence structure is broken down into the simplest (two-heavy-atom) molecules containing the same component bonds. 

P,T-PARITY VIOLATION EFFECTS IN POLAR HEAVY-ATOM MOLECULES... 

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

Study of P- and T-parity nonconservation effects in heavy-atom molecules Historical background... 

It is also critical to have a high value of the effective electric held IF, acting on the electron. The only way to know that parameter is to perform relativistic calculations. It is notable that the first semiempirical estimates of this kind were performed by Sandars in [16, 15] for Cs and TIF, correspondingly. In these papers the importance of accounting for relativistic effects and using heavy atoms and heavy-atom molecules in EDM experiments was first understood. 

The expected energy difference, 2d - E is extremely small even for completely polarized heavy-atom molecules. Thus, in practice, the EDM experiment is usually carried out in parallel and antiparallel electric and magnetic (B) fields. Interaction energy of the molecular magnetic moment, jl, with the magnetic field is much higher than that of the EDM with the electric field and the energy differences are... 

The most straightforward method for electronic structure calculation of heavy-atom molecules is solution of the eigenvalue problem using the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonians [4f, 42, 43] when some approximation for the four-component wave function is chosen. 

The shape-consistent (or norm-conserving ) RECP approaches are most widely employed in calculations of heavy-atom molecules though ener-gy-adjusted/consistent pseudopotentials [58] by Stuttgart team are also actively used as well as the Huzinaga-type ab initio model potentials [66]. In plane wave calculations of many-atom systems and in molecular dynamics, the separable pseudopotentials [61, 62, 63] are more popular now because they provide linear scaling of computational effort with the basis set size in contrast to the radially-local RECPs. The nonrelativistic shape-consistent effective core potential was first proposed by Durand Barthelat [71] and then a modified scheme of the pseudoorbital construction was suggested by Christiansen et al. [72] and by Hamann et al. [73]. 

The two-step method consists of a two-component molecular RECP calculation at the first step, followed by restoration of the proper four-component wave function in atomic cores at the second step. Though the method was developed originally for studying core properties in heavy-atom molecules, it can be efficiently applied to studying the properties described by the operators heavily concentrated in cores or on nuclei of light atoms in other computationally difficult cases, e.g., in many-atom molecules and solids. The details of these steps are described below. 

Generalized RECP When core electrons of a heavy-atom molecule do not play an active role, the effective Hamiltonian with RECP can be presented in the form... 

With the restored molecular bispinors, the two-electron integrals on them can be easily calculated. Thus, the four-component transfomation from the atomic basis that is a time-consuming stage of four-component calculations of heavy-atom molecules can be avoided. Besides, the VOCR technique developed in [92] for molecular pseudospinors can be reformulated for the case of molecular pseudospinorbitals to reduce the complexity of the molecular GRECP calculation as is discussed in section 5. 

The P,T-parity nonconservation parameters and hyperfine constants have been calculated for the particular heavy-atom molecules which are of primary interest for modern experiments to search for PNC effects. It is found that a high level of accounting for electron correlations is necessary for reliable calculation of these properties with the required accuracy. The applied two-step (GRECP/NOCR) scheme of calculation of the properties described by the operators heavily concentrated in atomic cores and on nuclei has proved to be a very efficient way to take account of these correlations with moderate efforts. The results of the two-step calculations for hyperfine constants differ by less than 10% from the corresponding exper-... 

We are very grateful to L.N. Labzowsky for initiating and supporting our activity in studying PNC properties in heavy-atom molecules over many years. We would like to thank also our colleagues I.V. Abarenkov, A.B. Alekseyev, R.J. Buenker, E. Eliav, U. Kaldor, M.G. Kozlov, A.I. Panin, A.V. Tulub, and LI. Tupitsyn for stimulating discussions and fruitful collaboration on the relevant research. 

An isodesmic reaction92 is a formal reaction, in which the number of electron pairs as well as formal chemical bond types are conserved while the relationships among the bonds are altered. A subclass of the isodesmic reactions is the class of bond separation energies, in which all formal bonds of a molecule are separated into two-heavy-atom molecules containing the same type of bonds. Stoichiometric balance is achieved for the bond separation energies by adding an appropriate number of one-heavy-atom hydrides to the left side of the reaction92. 

In this simple case there is no advantage to the pseudopotential calculation (the 3-21G( ) geometry is actually better ), but more challenging calculations on very-heavy-atom molecules, particularly transition metal molecules, rely heavily on ab initio or DFT (Chapter 7) calculations with pseudopotentials. Nevertheless, ordinary nonrelativistic all-electron basis sets sometimes give good results with quite heavy atoms [64]. A concise description of pseudopotential theory and specific relativistic effects on molecules, with several references, is given by Levine [65]. Reviews oriented toward transition metal molecules [66a,b,c] and the lanthanides [66d] have appeared, as well as detailed reviews of the more technical aspects of the theory [67]. See too Section 8.3. 

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