Fan Wang

On the relation between time-dependent and variational density functional theory approaches for the determination of excitation energies and transition moments.

It is shown that it is possible to derive the basic eigenvalue equation of adiabatic time-dependent density functional theory within the Tamm-Dancoff approximation (TD-DFT/TD) from a variational principle. The variational principle is applied to the regular Kohn-Sham formulation of DFT energy expression for a single Slater determinant and leads to the same energy spectrum as TD-DFT/TD. It is further shown that this variational approach affords the same electric and magnetic transition moments as TD-DFT/TD. The variational scheme can also be applied without the Tamm-Dancoff approximation. Practical implementations of TD-DFT are limited to second order response theory which introduces errors in transition energies for charge transfer and Rydberg excitations. It is indicated that higher order terms can be incorporated into the variational approach. It is also discussed how the current variational method is related to traditional DFT schemes based on variational principles such as DeltaSCF-DFT, and how they can be combined.

Structural and electronic properties of TaSin (n=1–13) clusters: A relativistic density functional investigation

The TaSi(n) (n=1-13) clusters with doublet, quartet, and sextet spin configurations have been systematically investigated by a relativistic density functional theory with the generalized gradient approximation available in Amsterdam density functional program. The total bonding energies, equilibrium geometries, Mulliken populations as well as Hirshfeld charges of TaSi(n) (n=1-13) clusters are calculated and presented. The emphasis on the stabilities and electronic properties is discussed. The most stable structures of the small TaSi(n) (n=1-6) clusters and the evolutional rule of low-lying geometries of the larger TaSi(n) (n=7-13) clusters are obtained. Theoretical results indicate that the most stable structure of TaSi(n) (n=1-6) clusters keeps the similar framework as the most stable structure of Si(n+1) clusters except for TaSi(3) cluster. The Ta atom in the lowest-energy TaSi(n) (n=1-13) isomers occupies a gradual sinking site, and the site moves from convex, to flatness, and to concave with the number of Si atom varying from 1 to 13. When n=12, the Ta atom in TaSi(12) cluster completely falls into the center of the Si frame, and a cagelike TaSi(12) geometry is formed. Meanwhile, the net Mulliken and Hirsheld populations of the Ta atom in the TaSi(n) (n=1-13) clusters vary from positive to negative, manifesting that the charges in TaSi(n) (n>/=12) clusters transfer from Si atoms to Ta atom. Additionally, the contribution of Si-Si and Si-Ta interactions to the stability of TaSi(n) clusters is briefly discussed. Furthermore, the investigations on atomic averaged binding energies and fragmentation energies show that the TaSi(n) (n=2,3,5,7,10,11,12) clusters have enhanced stabilities. Compared with pure silicon clusters, a universal narrowing of highest occupied molecular orbital-lowest unoccupied molecular orbital gap in TaSi(n) clusters is found.

Closed-shell coupled-cluster theory with spin-orbit coupling

A two-component closed-shell coupled-cluster (CC) approach using relativistic effective core potentials with spin-orbit coupling included in the post-Hartree-Fock treatment is proposed and implemented at the CC singles and doubles (CCSD) level as well as at the CCSD level augmented by a perturbative treatment of triple excitations [CCSD(T)]. The latter invokes as an additional approximation the neglect of the occupied-occupied and virtual-virtual blocks of the spin-orbit coupling matrix in order to avoid the iterative N(7) steps in the treatment of triple excitations. The computational effort of the implemented two-component CC methods is about 10-15 times that of its corresponding nonrelativistic counterpart, which needs to be compared to the by a factor of 32 higher cost for fully relativistic schemes and schemes with spin-orbit coupling included already at the Hartree-Fock self-consistent field (HF-SCF) level. This substantial computational saving is due to the use of real molecular orbitals and real two-electron integrals. Results on 5p-, 6p-, and 7p-block element compounds show that the bond lengths and harmonic frequencies obtained with the present two-component CCSD method agree well with those computed with the CCSD approach including spin-orbit coupling at the HF-SCF level even for the 7p-block element compounds. As for the CCSD(T) approach, high accuracy for 5p- and 6p-block element compounds is retained. However, the difference in bond lengths and harmonic frequencies becomes somewhat more pronounced for the 7p-block element compounds.

Spectroscopic constants of MH and M2 (M=Tl, E113, Bi, E115): Direct comparisons of four- and two-component approaches in the framework of relativistic density functional theory

The two-component DFT-ZORA (density functional theory, zeroth order regular approximation) method is implemented into the BDF (Beijing four-component density functional) program package so that systematic and direct comparisons between two- and four-component approaches are made possible for the first time. Different implementations of the ZORA method are also compared in this work. The calculated spectroscopic constants (bond lengths, binding energies, and force constants) for MH and M2 (M=Tl, E113, Bi, E115) by the two- and four-component approaches are very similar. The ionization and excitation energies for the metals obtained by these methods also agree very well with each other. Still, minor higher order relativistic effects beyond ZORA can be identified occasionally, but can be “safely” neglected. Therefore, the applicability of transformed (two-component) Hamiltonians to valence properties is well justified. However, the computational efficiency of four-component DFT compares favorably with that o...

Analytic energy gradients in closed-shell coupled-cluster theory with spin-orbit coupling

Gradients in closed-shell coupled-cluster (CC) theory with spin-orbit coupling included in the post Hartree-Fock treatment have been implemented at the CC singles and doubles (CCSD) level and at the CCSD level augmented by a perturbative treatment of triple excitations [CCSD(T)]. The additional computational effort required in analytic energy-gradient calculations is roughly the same as that for ground-state energy calculations in the case of CCSD, and it is about twice in the case of CCSD(T) calculations. The structures, harmonic frequencies, and dipole moments of some heavy-element compounds have been calculated using the present analytic energy-gradient techniques including spin-orbit coupling. The results show that spin-orbit coupling can have a significant influence on both the equilibrium structure and the harmonic vibrational frequencies and that its inclusion is essential to obtain reliable and accurate estimates for geometrical parameters of heavy-element compounds.

Symmetry exploitation in closed-shell coupled-cluster theory with spin-orbit coupling.

In the present work, we report exploitation of spatial symmetry in calculations of ground state energy and analytic first derivatives of closed-shell molecules based on our previously developed coupled-cluster (CC) approach with spin-orbit coupling. Both time-reversal symmetry and spatial symmetry for D(2h) and its subgroups are exploited in the implementation. The symmetry of a certain spin case for the amplitude, intermediate, or density matrix is determined by the symmetry of the corresponding spin functions and the direct product decomposition method is employed in computations involving these quantities. The reduction in computational effort achieved through the use of spatial symmetry is larger than the order of the molecular single point group. Symmetry exploitation renders application of the CC approaches with spin-orbit coupling to larger closed-shell molecules containing heavy elements with high accuracy.

Equation-of-Motion Coupled-Cluster Theory for Excitation Energies of Closed-Shell Systems with Spin–Orbit Coupling

Excitation energies of closed-shell systems based on the equation-of-motion (EOM) coupled-cluster theory at the singles and doubles (CCSD) level with spin-orbit coupling (SOC) included in the post-Hartree-Fock treatment are implemented in the present work. SOC can be included in both the CC and EOM steps (EOM-SOC-CCSD) or only in the EOM part (SOC-EOM-CCSD). The latter approach is an economical way to account for SOC effects, but excitation energies with this approach are not size-intensive. When the unlinked term in the latter approach is neglected (cSOC-EOM-CCSD), size-intensive excitation energies can be obtained. Time-reversal symmetry and spatial symmetry are exploited to reduce the computational effort. Imposing time-reversal symmetry results in a real matrix representation for the similarity-transformed Hamiltonian, which facilitates the requirement of time-reversal symmetry for new trial vectors in Davidsons algorithm. Results on some closed-shell atoms and molecules containing heavy elements show that EOM-SOC-CCSD can provide excitation energies and spin-orbit splittings with reasonable accuracy. On the other hand, the SOC-EOM-CCSD approach is able to afford accurate estimates of SOC effects for valence electrons of systems containing elements up to the fifth row, while cSOC-EOM-CCSD is less accurate for spin-orbit splittings of transitions involving p1/2 spinors, even for Kr.

Analytic second derivatives in closed-shell coupled-cluster theory with spin-orbit coupling

The theory for geometrical second derivatives of the energy is outlined for the recently suggested two-component coupled-cluster approach using relativistic effective core potentials with spin-orbit coupling included in the post-Hartree-Fock treatment [F. Wang, J. Gauss, and C. van Wullen, J. Chem. Phys. 129, 064113 (2008)], and an implementation is reported at the coupled-cluster singles and doubles (CCSD) level as well as at the CCSD level augmented by a perturbative treatment of triple excitations [CCSD(T)]. The applicability of the developed analytic second-derivative techniques is demonstrated by computing harmonic and fundamental frequencies for PtH(2), PbH(2), and HgH(2) with the required cubic and semidiagonal quartic force fields obtained by numerical differentiation of the analytically evaluated quadratic force constants. Spin-orbit coupling effects are shown to be non-negligible for the three considered molecules and thus need to be considered in the case of high-accuracy predictions.

Equation-of-motion coupled-cluster method for doubly ionized states with spin-orbit coupling

In this work, we report implementation of the equation-of-motion coupled-cluster method for doubly ionized states (EOM-DIP-CC) with spin-orbit coupling (SOC) using a closed-shell reference. Double ionization potentials (DIPs) are calculated in the space spanned by 2h and 3h1p determinants with the EOM-DIP-CC approach at the CC singles and doubles level (CCSD). Time-reversal symmetry together with spatial symmetry is exploited to reduce computational effort. To circumvent the problem of unstable dianion references when diffuse basis functions are included, nuclear charges are scaled. Effect of this stabilization potential on DIPs is estimated based on results from calculations using a small basis set without diffuse basis functions. DIPs and excitation energies of some low-lying states for a series of open-shell atoms and molecules containing heavy elements with two unpaired electrons have been calculated with the EOM-DIP-CCSD approach. Results show that this approach is able to afford a reliable description on SOC splitting. Furthermore, the EOM-DIP-CCSD approach is shown to provide reasonable excitation energies for systems with a dianion reference when diffuse basis functions are not employed.

Coupled-cluster method for open-shell heavy-element systems with spin-orbit coupling

The coupled-cluster approach with spin-orbit coupling (SOC) included in post-self-consistent field treatment (SOC-CC) using relativistic effective core potentials is extended to spatially non-degenerate open-shell systems in this work. The unrestricted Hartree-Fock determinant corresponding to the scalar relativistic Hamiltonian is employed as the reference and the open-shell SOC-CC approach is implemented at the CC singles and doubles (CCSD) level as well as at the CCSD level augmented by a perturbative treatment of triple excitations (CCSD(T)). Due to the breaking of time-reversal symmetry and spatial symmetry, this open-shell SOC-CC approach is rather expensive compared with the closed-shell SOC-CC approach. The open-shell SOC-CC approach is applied to some open-shell atoms and diatomic molecules with s1, p3, σ1, or π2 configuration. Our results indicate that rather accurate results can be achieved with the open-shell SOC-CCSD(T) approach for these systems. Dissociation energies for some closed-shell molecules containing heavy IIIA or VIIA atoms are also calculated using the closed-shell SOC-CC approach, where energies of the IIIA or VIIA atoms are obtained from those of the closed-shell ions and experimental ionization potentials or electron affinities. SOC-CCSD(T) approach affords reliable dissociation energies for these molecules. Furthermore, scalar-relativistic CCSD(T) approach with the same strategy can also provide reasonable dissociation energies for the 5th row IIIA or VIIA molecules, while the error becomes pronounced for the 6th row elements.