Development And Application Of Relativistic Coupled-Cluster Theory For Heavy Element

In quantum chemical calculations,relativistic effects are important in determining properties of heavy and superheavy elements.Relativistic effects include scalar relativistic effects and spin-orbit coupling（SOC）.Scalar relativistic effects can be calculated in a similar way as a non-relativistic electronic structure calculation when the untransformed two-electron interaction is employed,however,treatment of SOC is more complicated.The most rigorous way in dealing with relativistic effects is to solve the eigen equation of the four-component Dirac-Coulomb-Breit Hamiltonian.Computational cost of this approach is usually too large to be applied to larger systems.To reduce computational effort,some approximate relativistic Hamiltonians have been developed and the most popular method is the relativistic effective core potentials（RECPs）.On the other hand,in order to deal with electron-correlation effects and the excited states reliably,many quantum chemical methods have been developed,among which the coupled-cluster theory（CC）and equation-of-motion coupled-cluster theory（EOM-CC）are one of the most popular in these approaches.In EOM-CC calculations,the most rigorous way in dealing with SOC is to include SOC in self-consistent filed（SCF）calculations,and this method is termed as SOC-EOM-CC.Orbital relaxation effects due to SOC are fully considered in this method,but its calculation effort is rather large.To reduce calculation cost,CC and EOM-CC methods based on the closed-shell reference with SOC included in post-SCF calculations have been implemented previously in our group.EOM-CCSD methods with SOC for excitation energies（EEs）,ionization potentials（IPs）,electron affinities（EAs）,and double ionization potentials（DIPs）have been implemented,and they are termed as EOM-CCSD（SOC）.In the implementation,spatial and time-reversal symmetry as well as real spin-orbital are employed to reduce computational cost.Single excitation amplitudes in the CC theory can describe orbital relaxation reliably due to SOC,and this method can obtain highly accurate results even for closed-shell superheavy element molecules.To further reduce computation cost in treating SOC in EOM-CCSD calculations,it is possible to include SOC only in EOM-CCSD calculations with cluster amplitudes obtained from scalar relativistic CCSD calculations.This method is termed as EOM（SOC）-CCSD.Results of this method are not size-intensive.In EOM-CCSD calculation,the most economical method in dealing with SOC is the degenerate or quasi-degenerate perturbation method.In the third chapter,we report implementation of the EOM-CCSD method with a perturbative treatment of SOC,which is called p SOC-EOM-CCSD.In this method,SOC is only considered between degenerate or quasi-degenerate states obtained by scalar-relativistic EOM-CCSD.When considering SOC among all states,this method is equivalent to EOM（SOC）-CCSD and its results are thus not size-intensive.In addition,we also proposed rp SOC-EOM-CCSD method where only the right eigen-vectors in p SOC-EOM-CCSD are required.Compared with SOC-EOM-CCSD results,highly accurate results are achieved with EOM-CCSD（SOC）for elements up to the sixth-row elements and reasonable results are obtained for superheavy elements when contribution of p_1/2spinor to the involved occupied orbitals is insignificant.In all electron calculations,performance of EOM（SOC）-CCSD and（r）p SOC-EOM-CCSD is studied by comparing their results with those of EOM-CCSD（SOC）for elements down to the six-row.Our results show that EOM（SOC）-CCSD is not necessarily more accurate than the perturbative methods and should not be used in practical calculation.Perturbative methods can give reasonable results for elements up to the fifth-row,and the difference between results of p SOC-EOM-CCSD and rp SOC-EOM-CCSD is very small.rp SOC-EOM-CCSD is an economical and reliable method in dealing with SOC for these systems.In previous work,we has implemented EOM-CCSD（SOC）methods for EEs,IPs,EAs and DIPs.In the fourth chapter,we report implementation of EOM-CCSD（SOC）for double electron affinities（DEAs）,and this method is employed to investigate the properties of heavy elements with two unpaired electrons.Results of EOM-DEA-CCSD are more accurate than those of EOM-DIP-CCSD for atoms.However,the error of EOM-DEA-CCSD is larger than that of EOM-EE-CCSD for some molecules.Bond lengths for the ground state and some lowing-excited states of Ga H,In H and Tl H are underestimated pronouncedly,but reasonable EEs are still obtained.For the systems withπ²configuration,errors of splittings with EOM-DEA-CCSD are larger than that of EOM-DIP-CCSD,which may be related to larger maximum T₁and T₂amplitudes in the reference.To further improve accuracy of the EOM-CCSD method,the contribution of triple excitations must be considered.According to previous works,results of EOM-CCSD（T）（a）*with non-iterative triple excitations are in good agreement with those of CC3 with iterative triple excitations.However,EOM-CCSD（T）（a）*is implemented only for spin-singlet states,and EOM-CCSD（T）（a）*with SOC included in SCF is also reported.In the fifth chapter,we report implementations of EOM-CCSD（T）（a）*for EEs of both spin-singlet and spin-triplet states,and EOM-CCSD（T）（a）*with SOC included in CC calculations for EEs,IPs and EAs.To improve computation efficiency,we proposed r-EOM-CCSD（T）（a）*and l-EOM-CCSD（T）（a）*where only right-and left-vector are employed,respectively,as well as EOM-CCSD*where contribution of triple excitations is considered only for excited states.EEs of singlet/triplet valence and Rydberg states for some organic molecules are calculated with the above mentioned approaches.Our results show that EOM-CCSD（T）（a）*,r-EOM-CCSD（T）（a）*and l-EOM-CCSD（T）（a）*can provide reliable equilibrium structures and harmonic frequencies for excited states,compared with CC3 results.EOM-CCSD（T）（a）*and l-EOM-CCSD（T）（a）*can improve EEs with EOM-CCSD significantly,but the error of EEs with r-EOM-CCSD（T）（a）*is slightly larger.EOM-CCSD*can obtain reasonable structures and harmonic frequencies for excited states,but it underestimates EEs pronouncedly.In addition,performance of EOM-CCSD（T）（a）*with SOC on the properties of heavy element system is investigated.Our results show that EOM-CCSD（T）（a）*can provide reliable EEs,IPs and EAs,and their errors can be reduced pronouncedly.Reliable results can be obtained by l-EOM-CCSD（T）（a）*and r-EOM-CCSD（T）（a）*for the fourth-and fifth-row elements,but errors of l-EOM-CCSD（T）（a）*are larger than those of r-EOM-CCSD（T）（a）*for sixth-row elements with strong SOC.SO splittings with all methods are in good agreement with each other.Errors of EEs,IPs and EAs obtained by EOM-CCSD*are relatively large,although it can still obtain reasonable SO splittings due to systematic error cancellation.EOM-CC method can be applied to some systems with multireference states in a single reference framework.Cyclic C₃H₃is the simplest cyclic hydrocarbon,but its configuration is rather complicated.The ground ²E"state of the D_3hstructure distorts to C_3v,C_2v,C_sor C₂symmetry due to both Jahn-Teller（JT）effects and pseudo Jahn-Teller（PJT）effects.In order to clarify the characters of these stationary points on potential energy surface of this radical,CCSD,CCSD（T）,EOM-EA/IP-CCSD and density functional theory（DFT）are employed to study characters of these geometries.JT effects and PJT effects on the deformation between these stationary points are studied in detail.Results of these methods are consistent with each other except for the b₂mode of²B₁(C_2v)state.For this vibrational mode,the frequencies of CCSD and CCSD（T）are qualitatively incorrect,which may be related to the instability of their Hartree-Fock reference state.In DFT calculations,all exchange-correlation functionals can give reasonable results except for M06-2X.Our results show that the ²A’state of C_ssymmetry is the only minimum point on the potential energy surface of cyclic C₃H₃.The²A₂and²B₁states of the C_2vgeometry are second-order saddle points,while both the ²A state of the C₂structure and the ²A"state of the C_sstructure are transition states connecting global minima²A’state.