Relativistic effects, potential energy

Let us consider how independent /i(i ) 2 effects contribute to the v E) for the hydrogen halides, HX (X = I, Br, and Cl). The curves shown on Fig. 7.6 correspond to relativistic adiabatic potential energy curves (respectively 0 dotted, 0+ dashed, 1 and 2 solid) for HI obtained after diagonalization of the electronic plus spin-orbit Hamiltonians (see Section 3.1.2.2). The strong R-dependence of the electronic transition moment reflects the independence of the relative contributions of the case(a) A-S-Q basis states to each relativistic adiabatic II state. The independent experimental photodissociation cross sections are plotted as solid curves in Fig. 7.7 for HI and HBr. Note that, in addition to the independent variations in the A — S characters of each fl-state caused by All = 0 spin-orbit interactions, all transitions from the X1E+ state to states that dissociate to the X(2P) + H(2S) limit are forbidden in the separated atom limit because they are at best (2Pi/2 <— 2P3/2) parity forbidden electric dipole transitions on the X atom. In the case of the continuum region of an attractive potential, the energy dependence of the dissociation cross section exhibits continuity in the Franck-Condon factor density (see Fig. 7.18 Allison and Dalgarno, 1971 Smith, 1971 Allison and Stwalley, 1973). 

The cc-pVnZ(-PP) basis sets have the 10 core electrons of bromine replaced by a relativistic effective potential. At each level of theory are reported (a) AE based on electronic energies only (b) AE plus zero-point vibrational energy (ZPVE) corrections and (c) AE + ZPVE cor- ... 

Some scalar relativistic effects are included implicitly in calculations if pseudopotentials for heavy atoms are used to mimic the presence of core electrons there are several families of pseudopotentials available the effective core potentials (ECP) (Cundari and Stevens 1993 Hay and Wadt 1985 Kahn et al. 1976 Stevens et al. 1984), energy-adjusted pseudopotentials (Cao and Dolg 2006 Dolg 2000 Peterson 2003 Peterson et al. 2003), averaged relativistic effective potentials (AREP) (Hurley et al. 1986 Lajohn et al. 1987 Ross et al. 1990), model core potentials (MCP) (Klobukowski et al. 1999), and ab initio model potentials (AIMP) (Huzinaga et al. 1987). 

HF/CI calculations yielded De = 0.93eV (90 kJ/mol) relative to Pd atoms in their excited 4d 5s states (excitation energy 0.95 eV for each). 0 = 1.18 eV was obtained without correction for a basis set superposition error. for dissociation into ground-state atoms would thus become negative [1, 3]. Dq = 1.1 eV (106 kJ/mol) followed from an MP-LSD calculation [12]. Dissociation energies (of Fg, F, and two aa hole states) were based on MC-SCF calculations using a relativistic effective potential [5]. 

The Dirac operator incorporates relativistic effects for the kinetic energy. In order to describe atomic and molecular systems, the potential energy operator must also be modified. In non-relativistic theory the potential energy is given by the Coulomb operator. 

The total energy in ab initio theory is given relative to the separated particles, i.e. bare nuclei and electrons. The experimental value for an atom is the sum of all the ionization potentials for a molecule there are additional contributions from the molecular bonds and associated zero-point energies. The experimental value for the total energy of H2O is —76.480 a.u., and the estimated contribution from relativistic effects is —0.045 a.u. Including a mass correction of 0.0028 a.u. (a non-Bom-Oppenheimer effect which accounts for the difference between finite and infinite nuclear masses) allows the experimental non-relativistic energy to be estimated at —76.438 0.003 a.u. ... 

This expression is exact within our original approximation, where we have neglected relativistic effects of the electrons and the zero-point motions of the nuclei. The physical interpretation is simple the first term represents the repulsive Coulomb potential between the nuclei, the second the kinetic energy of the electronic cloud, the third the attractive Coulomb potential between the electrons and the nuclei, and the last term the repulsive Coulomb potential between the electrons. 

The CPF approach gives quantitative reement with the experimental spectroscopic constants (24-25) for the ground state of Cu2 when large one-particle basis sets are used, provided that relativistic effects are included and the 3d electrons are correlated. In addition, CPF calculations have given (26) a potential surface for Cus that confirms the Jahn-Teller stabilization energy and pseudorotational barrier deduced (27-28) from the Cus fluorescence spectra (29). The CPF method has been used (9) to study clusters of up to six aluminum atoms. 

Assuming that substituted Sb at the surface may work as catalytic active site as well as W, First-principles density functional theory (DFT) calculations were performed with Becke-Perdew [7, 9] functional to evaluate the binding energy between p-xylene and catalyst. Scalar relativistic effects were treated with the energy-consistent pseudo-potentials for W and Sb. However, the binding strength with p-xylene is much weaker for Sb (0.6 eV) than for W (2.4 eV), as shown in Fig. 4. 

Initially, the level of theory that provides accurate geometries and bond energies of TM compounds, yet allows calculations on medium-sized molecules to be performed with reasonable time and CPU resources, had to be determined. Systematic investigations of effective core potentials (ECPs) with different valence basis sets led us to propose a standard level of theory for calculations on TM elements, namely ECPs with valence basis sets of a DZP quality [9, 10]. The small-core ECPs by Hay and Wadt [11] has been chosen, where the original valence basis sets (55/5/N) were decontracted to (441/2111/N-11) withN = 5,4, and 3, for the first-, second-, and third-row TM elements, respectively. The ECPs of the second and third TM rows include scalar relativistic effects while the first-row ECPs are nonrelativistic [11], For main-group elements, either 6-31G(d) [12-16] all electron basis set or, for the heavier elements, ECPs with equivalent (31/31/1) valence basis sets [17] have been employed. This combination has become our standard basis set II, which is used in a majority of our calculations [18]. 

The only calculation we found for CdH is the work of Balasubramanian [68], using Cl with relativistic effective core potentials. The coupled-cluster results are presened in Table 6. Calculated values for R , cOg and Dg agree very well with experiment. Relativity contracts the bond by 0.04 and reduces the binding energy by 0.16 eV. The one- and two-component DK method reproduce the relativistic effects closely. Similar trends are observed for the excited states (Tables 7-9). Comparison with experiment is difficult for these states, since many of the experimental values are based on incomplete or uncertain data [65]. Calculated results for the CdH anion are shown in Table 10. The... 

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