Equations of explicitly-correlated coupled-cluster methods

The tensor contraction expressions defining a variety of high-rank coupled-cluster energies and wave functions that include the interelectronic distances (r(12)) explicitly (CC-R12) have been derived with the aid of a newly-developed computerized symbolic algebra smith. Efficient computational sequences to perform these tensor contractions have also been suggested, defining intermediate tensors-some reusable-as a sum of binary tensor contractions. smith can elucidate the index permutation symmetry of intermediate tensors that arise from a Slater-determinant expectation value of any number of excitation, deexcitation and other general second-quantized operators. smith also automates additional algebraic transformation steps specific to R12 methods, i.e. the identification and isolation of the special intermediates that need to be evaluated analytically and the resolution-of-the-identity insertion to facilitate high-dimensional molecular integral computation. The tensor contraction expressions defining the CC-R12 methods including through the connected quadruple excitation operator (CCSDTQ-R12) have been documented and efficient computational sequences have been suggested not just for the ground state but also for excited states via the equation-of-motion formalism (EOM-CC-R12) and for the so-called Lambda equation (Lambda-CC-R12) of the CC analytical gradient theory. Additional equations (the geminal amplitude equation) arise in CC-R12 that need to be solved to determine the coefficients multiplying the r(12)-dependent factors. The operation cost of solving the geminal amplitude equations of rank-k CC-R12 and EOM-CC-R12 (right-hand side) scales as O(n(6)) (k = 2) or O(n(7)) (k > or = 3) with the number of orbitals n and is surpassed by the cost of solving the usual amplitude equations O(n(2k+2)). While the complexity of the geminal amplitude equations of Lambda- and EOM-CC-R12 (left-hand side) nominally scales as O(n(2k+2)), it is less than that of the other O(n(2k+2)) terms in the usual amplitude equations. This suggests that the unabridged equations should be solved in high-rank CC-R12 for benchmark accuracy.     

Calculation of excited-state properties using general coupled-cluster and configuration-interaction models

Using string-based algorithms excitation energies and analytic first derivatives for excited states have been implemented for general coupled-cluster (CC) models within CC linear-response (LR) theory which is equivalent to the equation-of-motion (EOM) CC approach for these quantities. Transition moments between the ground and excited states are also considered in the framework of linear-response theory. The presented procedures are applicable to both single-reference-type and multireference-type CC wave functions independently of the excitation manifold constituting the cluster operator and the space in which the effective Hamiltonian is diagonalized. The performance of different LR-CC/EOM-CC and configuration-interaction approaches for excited states is compared. The effect of higher excitations on excited-state properties is demonstrated in benchmark calculations for NH(2) and NH(3). As a first application, the stationary points of the S(1) surface of acetylene are characterized by high-accuracy calculations.     

Spin-conserving and spin-flipping equation-of-motion coupled-cluster method with triple excitations

We report the implementation of the spin-conserving and spin-flipping variants of the equation-of-motion (EOM) coupled-cluster (CC) model, which includes single and double excitations in the CC part and single, double, and triple excitations in the EOM part, i.e., EOM-CC(2,3) [Hirata, Nooijen, Bartlett, Chem. Phys. Lett. 326, 255 (2000)] for closed- and open-shell references. Inclusion of triples significantly improves the accuracy of EOM-CCSD for excitation energies (EOM-EE-CCSD) and its spin-flip (SF) counterpart, EOM-SF-CCSD, especially when the reference wave function is strongly spin-contaminated. A less computationally demanding active space variant with semi-internal triples has also been implemented. The capabilities of full and active space EOM-CC(2,3) are demonstrated by applications to CO(+) and CH radicals as well as to the methylene and trimethylenemethane diradicals and the dehydro-m-xylylene triradical.     

Critical comparison of various connected quadruple excitation approximations in the coupled-cluster treatment of bond breaking

To assess the limits of single-reference coupled-cluster (CC) methods for potential-energy surfaces, several methods have been considered for the inclusion of connected quadruple excitations. Most are based upon the factorized inclusion of the connected quadruple contribution (Qf) [J. Chem. Phys. 108, 9221 (1998)]. We compare the methods for the treatment of potential-energy curves for small molecules. These include CCSD(TQf), where the initial contributions of triple (T) and factorized quadruple excitations are added to coupled-cluster singles (S) and doubles (D), its generalization to CCSD(TQf), where instead of measuring their first contribution from orders in H, it is measured from orders in H=e(-(T1+T2))He(T1+T2); renormalized approximations of both, and CCSD2 defined in [J. Chem. Phys. 115, 2014 (2001)]. We also consider CCSDT, CCSDT(Qf), CCSDTQ, and CCSDTQP for comparison, where T, Q, and P indicate full triple, quadruple, and pentuple excitations, respectively. Illustrations for F2, the double bond breaking in water, and N2 are shown, including effects of quadruples on equilibrium geometries and vibrational frequencies. Despite the fact that no perturbative approximation, as opposed to an iterative approximation, should be able to separate a molecule correctly for a restricted-Hartree-Fock reference function, some of these higher-order approximations have a role to play in developing new, more robust procedures.     

Method of moments of coupled-cluster equations: a new formalism for designing accurate electronic structure methods for ground and excited states

The method of moments of coupled-cluster equations (MMCC), which provides a systematic way of improving the results of the standard coupled-cluster (CC) and equation-of-motion CC (EOMCC) calculations for the ground- and excited-state energies of atomic and molecular systems, is described. The MMCC theory and its generalized MMCC (GMMCC) extension that enables one to use the cluster operators resulting from the standard as well as nonstandard CC calculations, including those obtained with the extended CC (ECC) approaches, are based on rigorous mathematical relationships that define the many-body structure of the differences between the full configuration interaction (CI) and CC or EOMCC energies. These relationships can be used to design the noniterative corrections to the CC/EOMCC energies that work for chemical bond breaking and potential energy surfaces of excited electronic states, including excited states dominated by double excitations, where the standard single-reference CC/EOMCC methods fail. Several MMCC and GMMCC approximations are discussed, including the renormalized and completely renormalized CC/EOMCC methods for closed- and open-shell states, the quadratic MMCC approaches, the CI-corrected MMCC methods, and the GMMCC approaches for multiple bond breaking based on the ECC cluster amplitudes.     

Higher-order equation-of-motion coupled-cluster methods for ionization processes

Compact algebraic equations defining the equation-of-motion coupled-cluster (EOM-CC) methods for ionization potentials (IP-EOM-CC) have been derived and computer implemented by virtue of a symbolic algebra system largely automating these processes. Models with connected cluster excitation operators truncated after double, triple, or quadruple level and with linear ionization operators truncated after two-hole-one-particle (2h1p), three-hole-two-particle (3h2p), or four-hole-three-particle (4h3p) level (abbreviated as IP-EOM-CCSD, CCSDT, and CCSDTQ, respectively) have been realized into parallel algorithms taking advantage of spin, spatial, and permutation symmetries with optimal size dependence of the computational costs. They are based on spin-orbital formalisms and can describe both alpha and beta ionizations from open-shell (doublet, triplet, etc.) reference states into ionized states with various spin magnetic quantum numbers. The application of these methods to Koopmans and satellite ionizations of N2 and CO (with the ambiguity due to finite basis sets eliminated by extrapolation) has shown that IP-EOM-CCSD frequently accounts for orbital relaxation inadequately and displays errors exceeding a couple of eV. However, these errors can be systematically reduced to tenths or even hundredths of an eV by IP-EOM-CCSDT or CCSDTQ. Comparison of spectroscopic parameters of the FH+ and NH+ radicals between IP-EOM-CC and experiments has also underscored the importance of higher-order IP-EOM-CC treatments. For instance, the harmonic frequencies of the A 2Sigma- state of NH+ are predicted to be 1285, 1723, and 1705 cm(-1) by IP-EOM-CCSD, CCSDT, and CCSDTQ, respectively, as compared to the observed value of 1707 cm(-1). The small adiabatic energy separation (observed 0.04 eV) between the X 2Pi and a 4Sigma- states of NH+ also requires IP-EOM-CCSDTQ for a quantitative prediction (0.06 eV) when the a 4Sigma- state has the low-spin magnetic quantum number (s(z) = 1/2). When the state with s(z) = 3/2 is sought, the energy separations converge much more rapidly with the IP-EOM-CCSD value (0.03 eV) already being close to the observed (0.04 eV).     

Making More Extensive Use of the Coupled-cluster Wave Function: from the Standard Energy Expression to the Energy Expectation Value

The standard coupled-cluster (CC) scheme with single and double excitations in the cluster operator (CCSD) includes only up to quadruple excitations in the equations. The CCSD exponential expansion generates, however, all possible excitations out of the reference function through products of the cluster operators. Clearly, in all standard approximate CC approaches only a part of the CC wave function is used in the equations. If the standard CCSD wave function is inserted into the energy expectation value expression then the complete CCSD wave function contributes to the energy. Such an energy expectation value expression can be presented as a sum of the standard CCSD energy formula plus correction terms. The correction terms provide an information about the quality of the total CC function. Contributions associated with the presence of higher than double excitations in the bra CCSD wave function supplement the CCSD energy obtained within the standard scheme. These contributions can be generated in a sequential way by considering intermediate excitation levels for the bra CCSD wave function in the expectation value expression before reaching the highest excitation level. In this way the importance of particular components differing in the standard and expectation value CCSD energies can be investigated. Some of the contributions can be recognized as close to or identical with the so-called renormalized noniterative corrections to the CC methods. We try to see to what an extent the nonstandard energy expressions, like the energy expectation value or the asymmetric energy formula, can be used to extend the applicability of the CCSD method illustrating our considerations with some numerical examples.Dedicated to Professor Jean-Paul Malrieu to honor his contribution to quantum chemistry and physics     

Charge-transfer separability and size-extensivity in the equation-of-motion coupled cluster method: EOM-CCx

We study the charge-transfer separability (CTS) property of the Fock space (FS) and equation-of-motion (EOM) coupled cluster (CC) methods by analysing the charge-transfer (CT) excitation energy versus the donor-acceptor (D-A) distance. All FS-CC approaches fulfill the CT separability condition which is not the case for the standard EOM-CC approaches. This defect of the EOM-CC scheme can be fixed by slight modification of the H matrix's diagrammatic structure, namely by adding some "dressing" composed of disconnected terms. The latter guarantee CTS of the respective EOM-CC scheme and marginally improve local excitations. The newly proposed variant of the EOM-CCSD approach is termed EOM-CCSDx (size-extensive EOM-CCSD).     

State-of-the-art computations of dipole moments using analytic gradients of high-level density-fitted coupled-cluster methods with focal-point approximations

Using the analytic derivatives approach, dipole moments of high-level density-fitted coupled-cluster (CC) methods, such as coupled-cluster singles and doubles (CCSD), and coupled-cluster singles and doubles with perturbative triples [CCSD(T)], are presented. To obtain the high accuracy results, the computed dipole moments are extrapolated to the complete basis set (CBS) limits applying focal-point approximations. Dipole moments of the CC methods considered are compared with the experimental gas-phase values, as well as with the common DFT functionals, such as B3LYP, BP86, M06-2X, and BLYP. For all test sets considered, the CCSD(T) method provides substantial improvements over Hartree–Fock (HF), by 0.076–0.213 D, and its mean absolute errors are lower than 0.06 D. Furthermore, our results indicate that even though the performances of the common DFT functionals considered are significantly better than that of HF, their results are not comparable with the CC methods. Our results demonstrate that the CCSD(T)/CBS level of theory provides highly-accurate dipole moments, and its quality approaching the experimental results. © 2019 Wiley Periodicals, Inc.     

Equation-of-motion coupled-cluster method for the study of shape resonance

The equation-of-motion coupled-cluster method (EOM-CC) is applied for the first time to calculate the energy and width of a shape resonance in an electron-molecule scattering. The procedure is based on inclusion of complex absorbing potential with EOM-CC theory. We have applied this method to investigate the shape resonance in e(-)N(2), e(-)CO, and e(-)C(2)H(2).     

Higher-order equation-of-motion coupled-cluster methods for electron attachment

High-order equation-of-motion coupled-cluster methods for electron attachment (EA-EOM-CC) have been implemented with the aid of the symbolic algebra program TCE into parallel computer programs. Two types of size-extensive truncation have been applied to the electron-attachment and cluster excitation operators: (1) the electron-attachment operator truncated after the 2p-1h, 3p-2h, or 4p-3h level in combination with the cluster excitation operator after doubles, triples, or quadruples, respectively, defining EA-EOM-CCSD, EA-EOM-CCSDT, or EA-EOM-CCSDTQ; (2) the combination of up to the 3p-2h electron-attachment operator and up to the double cluster excitation operator [EA-EOM-CCSD(3p-2h)] or up to 4p-3h and triples [EA-EOM-CCSDT(4p-3h)]. These methods, capable of handling electron attachment to open-shell molecules, have been applied to the electron affinities of NH and C2, the excitation energies of CH, and the spectroscopic constants of all these molecules with the errors due to basis sets of finite sizes removed by extrapolation. The differences in the electron affinities or excitation energies between EA-EOM-CCSD and experiment are frequently in excess of 2 eV for these molecules, which have severe multideterminant wave functions. Including higher-order operators, the EA-EOM-CC methods predict these quantities accurate to within 0.01 eV of experimental values. In particular, the 3p-2h electron-attachment and triple cluster excitation operators are significant for achieving this accuracy.     

High-order electron-correlation methods with scalar relativistic and spin-orbit corrections

An assortment of computer-generated, parallel-executable programs of ab initio electron-correlation methods has been fitted with the ability to use relativistic reference wave functions. This has been done on the basis of scalar relativistic and spin-orbit effective potentials and by allowing the computer-generated programs to handle complex-valued, spinless orbitals determined by these potentials. The electron-correlation methods that benefit from this extension are high-order coupled-cluster methods (up to quadruple excitation operators) for closed- and open-shell species, coupled-cluster methods for excited and ionized states (up to quadruples), second-order perturbation corrections to coupled-cluster methods (up to triples), high-order perturbation corrections to configuration-interaction singles, and active-space (multireference) coupled-cluster methods for the ground, excited, and ionized states (up to active-space quadruples). A subset of these methods is used jointly such that the dynamical correlation energies and scalar relativistic effects are computed by a lower-order electron-correlation method with more extensive basis sets and all-electron relativistic treatment, whereas the nondynamical correlation energies and spin-orbit effects are treated by a higher-order electron-correlation method with smaller basis sets and relativistic effective potentials. The authors demonstrate the utility and efficiency of this composite scheme in chemical simulation wherein the consideration of spin-orbit effects is essential: ionization energies of rare gases, spectroscopic constants of protonated rare gases, and photoelectron spectra of hydrogen halides.     

Application of equation-of-motion coupled-cluster methods to low-lying singlet and triplet electronic states of HBO and BOH

The equilibrium structures and physical properties of the X (1)sigma(+) linear electronic states, linear excited singlet and triplet electronic states of hydroboron monoxide (HBO) (A (1)sigma(-), B (1)delta, a (3)sigma(+), and b (3)delta) and boron hydroxide (BOH) (A (1)sigma(+), B (1)Pi, and b (3)Pi), and their bent counterparts (HBO a (3)A('), b (3)A("), A (1)A("), B (1)A(') and BOH X (1)A('), b (3)A('), c (3)A("), A (1)A('), B (1)A('), C (1)A(")) are investigated using excited electronic state ab initio equation-of-motion coupled-cluster (EOM-CC) methods. A new implementation of open-shell EOM-CC including iterative partial triple excitations (EOM-CC3) was tested. Coupled-cluster wave functions with single and double excitations (CCSD), single, double, and iterative partial triple excitations (CC3), and single, double, and full triple excitations (CCSDT) are employed with the correlation-consistent quadruple and quintuple zeta basis sets. The linear HBO X (1)sigma(+) state is predicted to lie 48.3 kcal mol(-1) (2.09 eV) lower in energy than the BOH X (1)sigma(+) linear stationary point at the CCSDT level of theory. The CCSDT BOH barrier to linearity is predicted to lie 3.7 kcal mol(-1) (0.16 eV). With a harmonic zero-point vibrational energy correction, the HBO X (1)sigma(+)-BOH X (1)A(') energy difference is 45.2 kcal mol(-1) (1.96 eV). The lowest triplet excited electronic state of HBO, a (3)A('), has a predicted excitation energy (T(e)) of 115 kcal mol(-1) (4.97 eV) from the HBO ground state minimum, while the lowest-bound BOH excited electronic state, b (3)A('), has a T(e) of 70.2 kcal mol(-1) (3.04 eV) with respect to BOH X (1)A('). The T(e) values predicted for the lowest singlet excited states are A (1)A(")<--X (1)sigma(+)=139 kcal mol(-1) (6.01 eV) for HBO and A (1)A(')<--X (1)A(')=102 kcal mol(-1) (4.42 eV) for BOH. Also for BOH, the triplet vertical transition energies are b (3)A(')<--X (1)A(')=71.4 kcal mol(-1) (3.10 eV) and c (3)A(")<--X (1)A(')=87.2 kcal mol(-1) (3.78 eV).     

Analytic second derivatives for general coupled-cluster and configuration-interaction models

Analytic second derivatives of energy for general coupled-cluster (CC) and configuration-interaction (CI) methods have been implemented using string-based many-body algorithms. Wave functions truncated at an arbitrary excitation level are considered. The presented method is applied to the calculation of CC and CI harmonic frequencies and nuclear magnetic resonance chemical shifts up to the full CI level for some selected systems. The present benchmarks underline the importance of higher excitations in high-accuracy calculations.     

Computerized implementation of higher‐order electron‐correlation methods and their linear‐scaling divide‐and‐conquer extensions

We have implemented a linear‐scaling divide‐and‐conquer (DC)‐based higher‐order coupled‐cluster (CC) and Møller–Plesset perturbation theories (MPPT) as well as their combinations automatically by means of the tensor contraction engine, which is a computerized symbolic algebra system. The DC‐based energy expressions of the standard CC and MPPT methods and the CC methods augmented with a perturbation correction were proposed for up to high excitation orders [e.g., CCSDTQ, MP4, and CCSD(2)_TQ]. The numerical assessment for hydrogen halide chains, polyene chains, and first coordination sphere (C1) model of photoactive yellow protein has revealed that the DC‐based correlation methods provide reliable correlation energies with significantly less computational cost than that of the conventional implementations. © 2017 Wiley Periodicals, Inc.     

Coupled-cluster methods including noniterative corrections for quadruple excitations

A new method is presented for treating the effects of quadruple excitations in coupled-cluster theory. In the approach, quadruple excitation contributions are computed from a formula based on a non-Hermitian perturbation theory analogous to that used previously to justify the usual noniterative triples correction used in the coupled cluster singles and doubles method with a perturbative treatment of the triple excitations (CCSD(T)). The method discussed in this paper plays a parallel role in improving energies obtained with the full coupled-cluster singles, doubles, and triples method (CCSDT) by adding a perturbative treatment of the quadruple excitations (CCSDT(Q)). The method is tested for an extensive set of examples, and is shown to provide total energies that compare favorably with those obtained with the full singles, doubles, triples, and quadruples (CCSDTQ) method.     

Theoretical study of electric field gradients at nitrogen nuclei in HNO,CH3NO and C2H3NO

Electric field gradients (EFGs) at the nitrogen nuclei of nitroxyl, nitrosomethane and nitrosoethylene were calculated by employing the complete-active-space self-consistent field (CASSCF), internally contracted multireference configuration interaction (icMRCI) and single-configuration coupled-cluster (CC) methods with correlation-consistent basis sets at the levels of attainable accuracy. Changes in the pσ and pπ atomic orbital populations were used to rationalize the differences between the N EFG tensor components related to the nitroso compound and separate nitric oxide. Calculated ¹⁴N nuclear quadrupole coupling constants were found in reasonable accord with experimental values. Comparison of electric dipole moments and potential energy characteristics with external values served to testify to good overall quality of the wave functions used in our calculations.     

Computational studies of photophysical properties of porphin, tetraphenylporphyrin and tetrabenzoporphyrin

The molecular photonics of porphyrins are studied using a combination of first-principle and semi-empirical calculations. The applicability of the approach is demonstrated by calculations on free-base porphyrin, tetraphenylporphyrin, and tetrabenzoporphyrin. The method uses excitation energies and oscillator strengths calculated at the linear-response time-dependent density functional theory (TDDFT) or the corresponding values calculated at the linear-response approximate second-order coupled-cluster (CC2) levels. The lowest singlet excitation energies obtained in the TDDFT and CC2 calculations are 0.0-0.28 eV and 0.18-0.47 eV larger than the experimental values, respectively. The excitation energies for the first triplet state calculated at the TDDFT level are in excellent agreement with experiment, whereas the corresponding CC2 values have larger deviations from experiment of 0.420.66 eV. The matrix elements of the spin-orbit and non-adiabatic coupling operators have been calculated at the semi-empirical intermediate neglect of differential overlap (INDO) level using a spectroscopic parameterization. The calculations yield rate constants for internal conversion and intersystem crossing processes as well as quantum yields for fluorescence and phosphorescence. The main mechanism for the quenching of fluorescence in tetraphenylporphyrin and tetrabenzoporphyrin is the internal conversion, whereas for free-base porphyrin both the internal conversion and the intersystem crossing processes reduce the fluorescence intensity. The phosphorescence is quenched by a fast internal conversion from the triplet to the ground state.     

On the accuracy of computed excited-state dipole moments

The dipole moments of furan and pyrrole in many electronically excited singlet states have been determined using coupled cluster theory including large one-electron basis sets. The inclusion of connected triple excitations is shown to uniformly decrease the equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) excitation energies by 0.04-0.24 eV, with an average reduction of 0.08 eV. Using a basis set larger than DZP (++)D (double-zeta plus polarization augmented with atom- and molecule-centered diffuse functions) uniformly increases the computed EOM-CCSD excitation energies by 0.03-0.29 eV, with an average increase of 0.20 eV. The corresponding shifts in excited-state dipole moments are more erratic. Including connected triple excitations changes the computed dipole moments by an rms amount of 0.17 au. More importantly, using a larger basis set shifts the dipole moments by an rms amount of 0.52 au, with an increase or a decrease being equally likely. The CC dipole moments are compared to those from time-dependent density functional theory (TD-DFT) computed by Burcl, Amos, and Handy [ Chem. Phys. Lett. 2002, 355, 8]. For 29 excited states of furan and pyrrole, the predicted TD-DFT dipole moments differ from the CC results by rms amounts of 1.6 au (HCTH functional) and 1.5 au (B97-1 functional). Including the asymptotic correction to TD-DFT developed by Tozer and Handy [ J. Chem. Phys. 1998, 109, 10180; J. Comput. Chem. 1999, 20, 106] reduces the rms differences for both functionals to 1.2 au. If those Rydberg excited states with very large polarizabilities are excluded, the rms differences from the CC results for the remaining 17 excited states become 1.31 au (HCTH) and 0.88 au (B97-1). For asymptotically corrected functionals and this subset of states, the rms differences from the CC results are only 0.54 au (HCTHc) and 0.34 au (B97-1c). Thus, the Tozer-Handy asymptotic correction for TD-DFT significantly improves the predictions of excited-state dipole moments. For excited states without very large polarizabilities, good agreement is achieved between excited-state dipole moments computed by coupled cluster theory and by the asymptotically corrected B97-1c density functional.