Second-order Correlation Corrections Research Articles

How do spin-scaled double hybrids designed for excitation energies perform for noncovalent excited-state interactions? An investigation on aromatic excimer models.

Time-dependent double hybrids with spin-component or spin-opposite scaling to their second-order perturbative correlation correction have demonstrated competitive robustness in the computation of electronic excitation energies. Some of the most robust are those recently published by our group (M. Casanova-Páez, L. Goerigk, J. Chem. Theory Comput. 2021, 20, 5165). So far, the implementation of these functionals has not allowed correctly calculating their ground-state total energies. Herein, we define their correct spin-scaled ground-state energy expressions which enables us to test our methods on the noncovalent excited-state interaction energies of four aromatic excimers. A range of 22 double hybrids with and without spin scaling are compared to the reasonably accurate wavefunction reference from our previous work (A. C. Hancock, L. Goerigk, RSC Adv. 2023, 13, 35964). The impact of spin scaling is highly dependent on the underlying functional expression, however, the smallest overall errors belong to spin-scaled functionals with range separation: SCS- and SOS- PBEPP86, and SCS-RSX-QIDH. We additionally determine parameters for DFT-D3(BJ)/D4 ground-state dispersion corrections of these functionals, which reduce errors in most cases. We highlight the necessity of dispersion corrections for even the most robust TD-DFT methods but also point out that ground-state based corrections are insufficient to completely capture dispersion effects for excited-state interaction energies.

Perspective on Simplified Quantum Chemistry Methods for Excited States and Response Properties.

We review recent developments in the framework of simplified quantum chemistry for excited state and optical response properties (sTD-DFT) and present future challenges for new method developments to improve accuracy and extend the range of application. In recent years, the scope of sTD-DFT was extended to molecular response calculations of the polarizability, optical rotation, first hyperpolarizability, two-photon absorption (2PA), and excited-state absorption for large systems with hundreds to thousands of atoms. The recently introduced spin-flip simplified time-dependent density functional theory (SF-sTD-DFT) variant enables an ultrafast treatment for diradicals and related strongly correlated systems. A few drawbacks were also identified, specifically for the computation of 2PA cross sections. We propose solutions to this problem and how to generally improve the accuracy of simplified schemes. New possible simplified schemes are also introduced for strongly correlated systems, e.g., with a second-order perturbative correlation correction. Interpretation tools that can extract chemical structure-property relationships from excited state or response calculations are also discussed. In particular, the recently introduced method-agnostic RespA approach based on natural response orbitals (NROs) as the key concept is employed.

Binding energies of the [formula omitted] states in boronlike argon

The binding energies of the ground 1s22s22p1/2 and first excited 1s22s22p3/2 states in boronlike argon are rigorously evaluated. The calculations are performed by the QED perturbation theory in the framework of the extended Furry picture taking into account all the relevant first- and second-order radiative and correlation corrections. The third- and higher-order interelectronic-interaction effects are considered within the Breit approximation. The relativistic nuclear recoil effect is taken into account. In comparison with the previous calculations of the binding energies in boronlike argon the accuracy of the theoretical predictions has been significantly improved.

Quantum chemical study of the reactions of Al, AlO and AlOH with H2O2

Quantum chemical calculations with the use of hybrid density functional with perturbative second-order correlation and dispersion correction are carried out to study the reactions of Al, AlO and AlOH with the H2O2 molecule. The values of energy barriers are estimated by means of extrapolation to the basis set limit. The energetically favorable reaction pathways have been revealed during the examination of the potential energy surfaces. Complex character of the processes under study has been established. It has been found that the initial stages of the considered processes are barrierless. Appropriate rate constants for principal channels of the reactions under study have been estimated by using capture model and canonical variational theory. The Arrhenius approximations for these channels have been proposed for the temperature range T=300–4000K.

MBPT calculation for the fine-structure intervals of principal series np(n=39) for Na

The fine-structure intervals of Na principal series np(n=39) are calculated by the many-body perturbation theory (MBPT) within the framework of relativity. To deal with the problem that a large set of continuum states is required in the MBPT calculation, an exponential potential is employed to generate a discrete, finite and nearly complete set of numerical basis functions. The zeroth-order wavefunctions and eignvalues are obtained by solving the relativistic Hartree-Fock (RHF) equation and the RHF equation with the action of a potential barrier. The basis set used in this work contains intermediate orbitals with the angular momentum l 6 and in an appropriate energy range, and most of them are the so called contracted orbitals. Encouraging results are obtained using this technique to calculate the second-order correlation corrections, combining the Breit effects in a first-order perturbation approach. Compared with other theoretical calculations, the present results are much close to the experimental results.

Correlation and relativistic effects in actinide ions

Wavelengths, line strengths, and transition rates are calculated for the multipole ($E1$, $M1$, $E2$, $M2$, $E3$, and $M3$) transitions between the excited $6{s}^{2}6{p}^{5}nl$ and $6s6{p}^{6}nl$ states and the ground $6{s}^{2}6{p}^{6}$ state in Ac${}^{3+}$, Th${}^{4+}$, and U${}^{6+}$ Rn-like ions. Relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate energies and transition rates for multipole transitions in these hole-particle systems. The RMBPT method agrees with multiconfigurational Dirac-Fock (MCDF) calculations in lowest order, includes all second-order correlation corrections, and includes corrections from negative-energy states. The calculations start from a [Xe]$4{f}^{14}5{d}^{10}6{s}^{2}6{p}^{6}$ Dirac-Fock potential. First-order perturbation theory is used to obtain intermediate-coupling coefficients, and second-order RMBPT is used to determine the matrix elements. Evaluated multipole matrix elements for transitions from excited states to the ground states are used to determine the line strengths, transition rates, and multipole polarizabilities. This work provides a number of yet unmeasured properties of these actinide ions for various applications and for benchmark tests of theory and experiment.

Correlation and relativistic effects for the4f−nland5p−nlmultipole transitions in Er-like tungsten

Wavelengths, transition rates, and line strengths are calculated for the multipole ($E1$, $M1$, $E2$, $M2$, $E3$, and $M3$) transitions between the excited [Cd]$4{f}^{13}5{p}^{6}\mathit{nl}$, [Cd]$4{f}^{14}5{p}^{5}\mathit{nl}$ configurations and the ground [Cd]$4{f}^{14}5{p}^{6}$ state in Er-like ${\mathrm{W}}^{6+}$ ion ([Cd]=[Kr]$4{d}^{10}5{s}^{2}$). In particular, the relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate energies and transition rates for multipole transitions in this hole-particle system. This method is based on the relativistic many-body perturbation theory that agrees with multiconfiguration Dirac-Fock (MCDF) calculations in lowest order, and includes all second-order correlation corrections and corrections from negative-energy states. The calculations start from a [Cd]$4{f}^{14}5{p}^{6}$ Dirac-Fock (DF) potential. First-order perturbation theory is used to obtain intermediate-coupling coefficients, and second-order RMBPT is used to determine the multipole matrix elements needed for calculations of other atomic properties such as line strengths and transition rates. In addition, core multipole polarizability is evaluated in random-phase and DF approximations. The comparison with available data is demonstrated.

Correlation and relativistic effects for the4f−nlmultipole transitions in Yb III ions

Wavelengths, transition rates, and line strengths are calculated for the multipole (E1, M1, E2, M2, and E3) transitions between the excited $[\mathrm{Xe}]4{f}^{13}ns$, $[\mathrm{Xe}]4{f}^{13}nd$, and $[\mathrm{Xe}]4{f}^{13}np$ and the ground $[\mathrm{Xe}]4{f}^{14}$ state in Yb III ion with the nuclear charge $Z=70$ $([\mathrm{Xe}]=[\mathrm{Ni}]4{s}^{2}4{p}^{6}4{d}^{10}5{s}^{2}5{p}^{6})$. Relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate energies and transition rates for multipole transitions in this hole-particle system. This method is based on the relativistic many-body perturbation theory that agrees with multiconfiguration Dirac-Fock calculations in lowest-order, includes all second-order correlation corrections, and includes corrections from negative energy states. The calculations start from a $[\mathrm{Xe}]4{d}^{14}$ Dirac-Fock potential. First-order perturbation theory is used to obtain intermediate-coupling coefficients, and second-order RMBPT is used to determine the matrix elements. Evaluated multipole matrix elements for transitions from excited states into the ground states and transitions between excited states are used to determine the lifetime of the 20 $[\mathrm{Xe}]4{f}^{13}5d(J)$ levels, four $[\mathrm{Xe}]4{f}^{13}6s(J)$ levels, 12 $[\mathrm{Xe}]4{f}^{13}6p(J)$ levels, and four $[\mathrm{Xe}]4{f}^{13}7s(J)$ levels.

Relativistic many-body calculations of atomic properties in Pd-like ions

Wavelengths, transition rates, and line strengths are calculated for the 85 possible multipole transitions between the excited 4p6 4d9 4f, 4p6 4d9 5l, 4p5 4d10 4f, and 4p5 4d10 5l states and the ground 4p6 4d10 state in Pd-like ions with the nuclear charges ranging from Z = 47 to 100. Relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate energies and transition rates for multipole transitions in hole–particle systems. This method is based on the relativistic many-body perturbation theory, agrees with MCDF calculations in lowest order, includes all second-order correlation corrections, and includes corrections from negative energy states. The calculations start from a [Zn]4p64d10 Dirac–Fock potential. First-order perturbation theory is used to obtain intermediate-coupling coefficients, and second-order RMBPT is used to determine the matrix elements. The contributions from negative-energy states are included in the second-order multipole matrix elements. The resulting transition energies and transition rates are compared with experimental values and with results from other recent calculations. Trends of the transitions rates for the selected multipole transitions as function of Z are illustrated graphically. The Z dependence of the energy splitting for all triplet terms of the 4p64d9 4f and 4p64d9 5l configurations are shown for Z = 47–100. PACS Nos.: 31.15.Ar, 31.15.Md, 32.70.Cs, 32.30.Rj, 31.25.Jf

Hartree–Fock studies of quantum dots and the 0.7 anomaly

We apply unrestricted Hartree–Fock to modelling two systems: (1) We calculate the spin structure and addition spectra of small symmetric quantum dots (often called 2D “artificial atoms”), improving the accuracy considerably by including, for the first time, second-order correlation corrections. We compare the results to experiment and to previous numerical works, and find that our spin structure in some cases disagrees with that calculated within mean-field theories, such as Hartree–Fock without correlation corrections, or density-functional theory [C. Sloggett, O.P. Sushkov, Phys. Rev. B 71 (2005) 235326]. (2) We model the well-known 0.7 anomaly in the conductance of a quantum point contact. We calculate the conductance using direct calculation of scattering phases on a ring, within Hartree–Fock. We observe strong localisation of the Fermi electrons on the barrier, and suggest a mechanism for the observed temperature-dependent conductance anomaly.

Multipole (E1, M1, E2, M2, E3, M3) transition wavelengths and rates between 3l−15l′ excited and ground states in nickel-like ions

A relativistic many-body method is used to calculate energy and transition rates for multipole transitions in many-electron ions. This method is based on relativistic many-body perturbation theory (RMBPT), agrees with MCDF calculations in lowest order, includes all second-order correlation corrections and includes corrections from negative energy states. Reduced matrix elements, oscillator strengths and transition rates are calculated for electric-multipole (dipole (E1), quadrupole (E2) and octupole (E3)) and magnetic-multipole (dipole (M1), quadrupole (M2) and octupole (M3)) transitions between 3l−15l′ excited and ground states in Ni-like ions with nuclear charges ranging from Z = 30 to 100. The calculations start from a 1s22s22p63s23p63d10 Dirac–Fock potential. First-order perturbation theory is used to obtain intermediate-coupling coefficients, and second-order RMBPT is used to determine the matrix elements. A detailed discussion of the various contributions to the dipole matrix elements and energy levels is given for nickel-like tungsten (Z = 74). The contributions from negative-energy states are included in the second-order E1, M1, E2 M2, E3 and M3 matrix elements. The resulting transition energies and transition rates are compared with experimental values and with results from other recent calculations. These atomic data are important in modelling of M-shell radiation spectra of heavy ions generated in electron beam ion trap experiments and in M-shell diagnostics of plasmas.

Relativistic many-body calculations of multipole (E1, M1, E2, M2, E3, and M3) transition wavelengths and rates between 3l −14l′ excited and ground states in nickel-like ions

Wavelengths, transition rates, and line strengths are calculated for the 76 possible multipole (E1, M1, E2, M2, E3, and M3) transitions between the excited 3s 23p 63d 94l, 3s 23p 53d 104l, and 3s3p 63d 104l and the ground 3s 23p 63d 10 states in Ni-like ions with the nuclear charges ranging from Z = 30 to 100. The relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate energies and transition rates for multipole transitions in hole–particle systems. This method is based on relativistic many-body perturbation theory, agrees with MCDF calculations in lowest-order, includes all second-order correlation corrections, and includes corrections from negative energy states. The calculations start from a 1s 22s 22p 63s 23p 63d 10 Dirac–Fock potential. First-order perturbation theory is used to obtain intermediate-coupling coefficients, and the second-order RMBPT is used to determine the matrix elements. The contributions from negative-energy states are included in the second-order E1, M1, E2, M2, E3, and M3 matrix elements. The resulting transition energies and transition rates are compared with experimental values and with results from other recent calculations. As a result, we present wavelengths and transition rates data for the selected transitions that include the 76 possible multipole (E1, M1, E2, M2, E3, and M3) transitions between the excited 3s 23p 63d 94l, 3s 23p 53d 104l, and 3s3p 63d 104l states and the ground 3s 23p 63d 10 state in Ni-like ions. Trends of the line strengths for the 76 multipole transitions and oscillator strengths for the 13 E1 transitions as function of Z are illustrated graphically. The Z-dependence of the energy splitting for all triplet terms of the 3s 23p 63d 94l, 3s 23p 53d 104l, and 3s3p 63d 104l configurations are shown in the range of Z = 30–100.

E1, E2, M1, and M2 transitions in the nickel isoelectronicsequence

A relativistic many-body method is developed to calculate energy and transition rates for multipole transitions in many-electron ions. This method is based on relativistic many-body perturbation theory (RMBPT), agrees with MCDF calculations in lowest order, includes all second-order correlation corrections, and includes corrections from negative-energy states. Reduced matrix elements, oscillator strengths, and transition rates are calculated for electric-dipole (E1) and electric-quadrupole (E2) transitions, and magnetic-dipole (M1) and magnetic-quadrupole (M2) transitions in Ni-like ions with nuclear charges ranging from Z = 30 to 100. The calculations start from a 1s22s22p63s23p63d10 Dirac–Fock potential. First-order perturbation theory is used to obtain intermediate-coupling coefficients, and second-order RMBPT is used to determine the matrix elements. The contributions from negative-energy states are included in the second-order E1, M1, E2, and M2 matrix elements. The resulting transition energies and transition rates are compared with experimental values and withresults from other recent calculations.PACS Nos.: 32.30.Rj, 32.70.Cs, 32.80.Rm, 34.70.+e

Basis set effects on the intermolecular interaction of the CF 4–CF 4 system

The intermolecular potential energy surface of the CF 4–CF 4 system is explored by means of ab initio molecular orbital calculations at the HF and MP2 level of theory. A number of progressively larger basis sets (up to 6-311+G(3d)) are used in order to evaluate the basis set effect. The HF energy component of the total interaction energy is somewhat affected by the choice of the basis set. The second-order correlation correction E(2) to the interaction energy is greatly changed with the choice of the basis set. The nonbonding parameters of the Lennard–Jones and the Buckingham potential energy functions are fitted to the ab initio MP2/6-31+G(3d) interaction energies for nine different orientations and a total of 130 configurations.

Calculations of magnetic properties. II. Electron-correlated nuclear shielding constants for nine small molecules

Calculations of the nuclear magnetic shielding constants at the self-consistent-field (SCF) and second-order Mo/ller–Plesset (MP2) levels of theory are reported for H2, N2, F2, HF, CO, HCN, HNC, H2O, and NH3. The reliability of the calculations is verified by the high degree of satisfaction of the appropriate sum rule. The second-order correlation corrections to the shielding constants are found to be particularly important for the multiply bonded atoms in molecules such as N2, CO, HCN, and HNC. Vibrational corrections to the shielding constants for the diatomic molecules are shown to be significant.

Multiparticle perturbation theory and an accurate calculation of parity nonconservation in cesium

A method of calculating higher-order correlation corrections, using Green's functions and the Feynman diagrammatic technique, is developed. A basis of the single-electron orbitals is computed using the relativistic Hartree-Fock method. The interaction of an atom with an external field is computed by solving the time-dependent Hartree-Fock equations. In the methodology presented, we consider all the second-order correlation corrections and the dominating classes of higher order diagrams: the screening of the Coulomb interaction of electrons, particle-hole interaction and mass operator iterations. The calculation of the energy levels, the intervals of hyperfine structure and the amplitudes of the allowed EI-transitions in Cs shows that the method developed ensures precision at the 0.1–1% level. A calculation of the parity nonconservation of the El-amplitude of the transition 6s–7s in Cs is produced. The result = −(0.91 ± 0.01)·10−11 i¦e¦aB (−QW/N) is obtained.

Comparison of Morokuma and perturbation theory approaches to decomposition of interaction energy. (NH 4) +…NH 3

The interaction energy between NH 4 + and NH 3 is decomposed for a range of intermolecular separations within the context of a variety of basis sets including 6-31G **, 6-31+G **, and Sadlej polarization sets of type [5s3p2d/3s2p], including a set of f-functions as well. The electrostatic component dominates the interaction in all cases. At separations near the minimum or closer, the Morokuma second-order components (POL and CT) blow up. In contrast, Δ E SCF def, representing mutual polarization and orbital relaxation, is in good agreement with the induction energy computed via perturbation theory. Correction of the superposition error is noted to be important for this agreement and, indeed, for all components. In fact, following this correction, even the smaller sets provide accurate measures of the exchange term. The second-order correlation correction to the electrostatic interaction ϵ es 12 is computed and found to be fairly small. The dispersion energy embodied in ϵ disp 20 is considerably larger than the latter term and grows in proportion to the flexibility of the basis set.

Analysis of the potential energy surface of Ar–NH3

The combination of supermolecular Mo/ller–Plesset treatment with the perturbation theory of intermolecular forces is applied in the analysis of the potential energy surface of Ar–NH3. Anisotropy of the self-consistent field (SCF) potential is determined by the first-order exchange repulsion. Second-order dispersion energy, the dominating attractive contribution, is anisotropic in the reciprocal sense to the first-order exchange, i.e., minima in one nearly coincide with maxima in the other. The estimated second-order correlation correction to the exchange effect is nearly as large as a half ΔESCF in the minimum and has a ‘‘smoothing’’ effect on the anisotropy of ε(20)disp. The model which combines ΔESCF with dispersion energy (SCF+D) is not accurate enough to quantitatively describe both radial and angular dependence of interaction energy. Comparison is also made between Ar–NH3 and Ar–PH3, as well as with the Ar dimer.

Many-body perturbation-theory calculations of energy levels along the lithium isoelectronic sequence.

Energies of n=2 states for ions of the lithium isoelectronic sequence are calculated from Z=3--92, starting from a Hartree-Fock potential and including second- and third-order correlation corrections, the lowest-order Breit interaction with retardation treated exactly, the second-order correlation corrections to the Breit interaction, and corrections for reduced mass and mass polarization. The resulting differences between theory and experiment for the 2p fine structure and the 2s-2p splittings are found to be in rough agreement with the one-electron Lamb shift, but clear deviations can be seen. A discussion is given of the calculations required to evaluate these deviations within the framework of quantum electrodynamics.

On the connection between the supermolecular Møller-Plesset treatment of the interaction energy and the perturbation theory of intermolecular forces

This paper presents the first rigorous analysis of the MP2 and MP3 supermolecular treatments in terms of the perturbation theory of intermolecular forces. In order to connect the two formalisms the MP energies are first expanded in terms of an auxiliary double-perturbation theory in spirit similar to the one proposed by Sadlej (1980, Molec. Phys., 39, 1249). In the next step, each term of this expansion is related to the perturbation theory of intermolecular interactions in the formulation of Szalewicz and Jeziorski (1979, Molec. Phys., 38, 191). Although the formal analysis neglects intermolecular exchange effects, a generalization of the final results is proposed so as to accommodate exchange effects. The supermolecular MP2 interaction energy contains the second-order intrasystem correlation correction to the Coulomb energy, in addition to the UCHF dispersion. The supermolecular MP3 term additionally involves the following: first-order intrasystem (apparent) correlation correction to the second-order UC...