Analytic calculation of second-order electric response properties with the normalized elimination of the small component (NESC) method

Analytic second derivatives of the relativistic energy for the calculation of electric response properties are derived utilizing the normalized elimination of the small component (NESC) method. Explicit formulas are given for electric static dipole polarizabilities and infrared intensities by starting at the NESC representation of electric dipole moments. The analytic derivatives are implemented in an existing NESC program and applied to calculate dipole moments, polarizabilities, and the infrared spectra of gold- and mercury-containing molecules as well as some actinide molecules. Comparison with experiment reveals the accuracy of NESC second order electric response properties.     

Relativistic calculation of nuclear magnetic shielding using normalized elimination of the small component

The normalized elimination of the small component (NESC) theory, recently proposed by Filatov and Cremer, is extended to include magnetic interactions and applied to the calculation of the nuclear magnetic shielding in HX (X=F, Cl, Br, I) systems. The NESC calculations are performed at the levels of the zeroth-order regular approximation (ZORA) and the second-order regular approximation (SORA). The calculations show that the NESC-ZORA results are very close to the NESC-SORA results, except for the shielding of the I nucleus. Both the NESC-ZORA and NESC-SORA calculations yield very similar results to the previously reported values obtained using the relativistic infinite-order two-component coupled Hartree-Fock method. The difference between NESC-ZORA and NESC-SORA results is significant for the shieldings of iodine.     

Connection between the regular approximation and the normalized elimination of the small component in relativistic quantum theory

The regular approximation to the normalized elimination of the small component (NESC) in the modified Dirac equation has been developed and presented in matrix form. The matrix form of the infinite-order regular approximation (IORA) expressions, obtained in [Filatov and Cremer, J. Chem. Phys. 118, 6741 (2003)] using the resolution of the identity, is the exact matrix representation and corresponds to the zeroth-order regular approximation to NESC (NESC-ZORA). Because IORA (=NESC-ZORA) is a variationally stable method, it was used as a suitable starting point for the development of the second-order regular approximation to NESC (NESC-SORA). As shown for hydrogenlike ions, NESC-SORA energies are closer to the exact Dirac energies than the energies from the fifth-order Douglas-Kroll approximation, which is much more computationally demanding than NESC-SORA. For the application of IORA (=NESC-ZORA) and NESC-SORA to many-electron systems, the number of the two-electron integrals that need to be evaluated (identical to the number of the two-electron integrals of a full Dirac-Hartree-Fock calculation) was drastically reduced by using the resolution of the identity technique. An approximation was derived, which requires only the two-electron integrals of a nonrelativistic calculation. The accuracy of this approach was demonstrated for heliumlike ions. The total energy based on the approximate integrals deviates from the energy calculated with the exact integrals by less than 5 x 10(-9) hartree units. NESC-ZORA and NESC-SORA can easily be implemented in any nonrelativistic quantum chemical program. Their application is comparable in cost with that of nonrelativistic methods. The methods can be run with density functional theory and any wave function method. NESC-SORA has the advantage that it does not imply a picture change.     

Density-functional and electron correlated study of five linear birefringences--Kerr, Cotton-Mouton, Buckingham, Jones, and magnetoelectric--in gaseous benzene

We present the results of an extended study of five birefringences--Kerr, Cotton-Mouton, Buckingham, Jones, and Magnetoelectric--on benzene in the gas phase. The relevant molecular quantities--first-order properties, linear, quadratic, and cubic response functions--are computed employing the density-functional theory (DFT) response theory, with a choice of functionals. In some cases, different functionals are employed for the wave-function computational step and for the subsequent analytical response calculation to determine the combination yielding at the same time the optimal energy and energy derivative results. Augmented correlation consistent basis sets of double and triple zeta quality are used. The DFT results are compared to those obtained at the Hartree-Fock level and in some cases within a coupled cluster singles and doubles electronic structure model. The study tries to assess the ability of the DFT response theory to describe a wide range of properties in a system of rather large size and high complexity. The relative strength of the five birefringences for plausible experimental conditions is determined and, when possible, comparison is made with the results of the measurements.     

An improved algorithm for the normalized elimination of the small-component method

A new algorithm for the iterative solution of the normalized elimination of the small component (NESC) method is presented that is less costly than previous algorithms and that is based on (1) solving the NESC equations for the uncontracted rather than contracted basis (??First-Diagonalize-then-Contract??), (2) a new iterative procedure for obtaining the NESC Hamiltonian (??iterative TU algorithm??), (3) the renormalization scheme connected to the picture change, and (4) a finite nucleus model with a Gaussian charge distribution. The accuracy of NESC energies, which match those of 4-component Dirac calculations, is demonstrated. Test calculations with CCSD(T), DFT, and large basis sets including high angular momentum basis functions (f,g,h,i) are presented to prove the general applicability of the new NESC algorithm. Comparison with other algorithms of solving the NESC equations are shortly discussed and time savings are presented.     

Wave‐function frozen‐density embedding: Approximate analytical nuclear ground‐state gradients

We report the derivation of approximate analytical nuclear ground‐state uncoupled frozen density embedding (FDEu) gradients for the resolution of identity (RI) variant of the second‐order approximate coupled cluster singles and doubles (RICC2) as well as density functional theory (DFT), and an efficient implementation thereof in the KOALA program. In order to guarantee a computationally efficient treatment, those gradient terms are neglected which would require the exchange of orbital information. This approach allows for geometry optimizations of single molecules surrounded by numerous molecules with fixed nuclei at RICC2‐in‐RICC2, RICC2‐in‐DFT, and DFT‐in‐DFT FDE level of theory using a dispersion correction, required due to the DFT‐based treatment of the interaction in FDE theory. Accuracy and applicability are assessed by the example of two case studies: (a) the Watson‐Crick pair adenine‐thymine, for which the optimized structures exhibit a maximum error of about 0.08 Å for our best scheme compared to supermolecular reference calculations, (b) carbon monoxide on a magnesium oxide surface model, for which the error amount up to 0.1 Å for our best scheme. Efficiency is demonstrated by successively including environment molecules and comparing to an optimized conventional supermolecular implementation, showing that the method is able to outperform conventional RICC2 schemes already with a rather small number of environment molecules, gaining significant speed up in computation time. © 2016 Wiley Periodicals, Inc.     

Atomistic Model of Uranium

The electronic state and potential data of U₂ molecules are performed by first principle calculations with B3LYP hybrid exchange-correlation functional, the valence electrons of U atom are treated with the (5s4p3d4f)/[3s3p2d2f] contraction basis sets, and the cores are approximated with the relativistic effective core potential. The results show that the ground electronic state is X⁹∑⁺_g . The pair potential data are fitted with a Murrell-Sorbie analytical potential function. The U-U embedded atom method (EAM) interatomic potential is deter-mined based on the generalized gradient approximation calculation within the framework of the density functional theory using Perdew-Burke-Ernzerhof exchange-correlation functional at the spin-polarized level. The physical properties, such as the cohesive energy, the lattice constant, the bulk modulus, the shear modulus, the sc/fcc relative energy, the hcp/fcc rela-tive energy, the shear modulus and the monovacancy formation energy are used to evaluate the EAM potential parameters. The U-U pair potential determined by the first principle calculations is in agreement with that defined by the EAM potential parameters. The EAM calculated formation energy of the monovacancy in the fcc structure is also found to be in close agreement with DFT calculation.     

Molecular-orbital-free algorithm for the excited-state force in time-dependent density functional theory

Starting from the equation of motion in the density matrix formulation, we reformulate the analytical gradient of the excited-state energy at the time-dependent density functional theory level in the nonorthogonal Gaussian atom-centered orbital (AO) basis. Analogous to the analytical first derivative in molecular-orbital (MO) basis, a Z-vector equation has been derived with respect to the reduced one-electronic density matrix in AO basis, which provides a potential possibility to exploit quantum locality of the density matrix and avoids the matrix transformation between the AO and the MO basis. Numerical tests are finished for the excited-state geometry optimization and adiabatic excitation energy calculation of a series of small molecules. The results demonstrate the computational efficiency and accuracy of the current AO-based energy gradient expression in comparison with the MO-based scheme.     

The generation of ground-state structures and electronic properties of ternary AlkTilNim clusters (k + l + m = 4) from a two-stage density functional theory global searching approach

Structural and electronic properties of ternary clusters Al_kTi_lNi_m, where k, l, and m are integers and k + l + m = 4 , are investigated. These clusters are generated and studied by performing a two-stage density functional theory (DFT) calculations using the Slater, Vosko, Wilks, and Nusair (SVWN) and Becke three-parameter, Lee-Yang-Parr (B3LYP) functional exchange correlations. In the first stage, an unbiased global search algorithm coupled with a DFT code with a light exchange-correlation and smaller basis sets are used to generate the lowest energy cluster structures. It is then followed by further optimization using another round of DFT calculation with heavy exchanged correlations and large basis set. Electronic properties of the structures obtained via the two-stage procedure are then studied via DFT calculations. The results are illustrated in the form of ternary diagram. Our DFT calculations find that the stability of the cluster increases with the increase in the number of nickel atoms inside the clusters. Our findings provide new insight into the ternary metallic cluster through the structure, stability, chemical order, and electronic properties studies.     

Generation of ground-state structures and electronic properties of ternary AlxTiyNiz clusters (x + y + z = 6) with a two-stage density functional theory global search approach

The structural and electronic properties of ternary Al_xTi_yNi_z clusters, where x, y, and z are integers and x + y + z = 6 , are investigated. Both Slater, Vosko, Wilks, and Nusair and B3LYP exchange-correlation (XC) functionals are employed in a two-stage density functional theory (DFT) calculations to generate these clusters. In the first stage, a minimum energy cluster structure is generated by an unbiased global search algorithm coupled with a DFT code using a light XC functional and small basis sets. In the second stage, the obtained cluster structure is further optimized by another round of global minimization search coupled with a DFT calculator using a heavier XC functional and more costly basis set. Electronic properties of the structures are illustrated in the form of a ternary diagram. Our DFT calculations find that the thermodynamic stability of the clusters increases with the increment in the number of constituent nickel atoms. These results provide a new insight to the structure, stability, chemical order, and electronic properties for the ternary alloy nanoclusters.     

Excited state geometry optimizations by analytical energy gradient of long-range corrected time-dependent density functional theory

An analytical excitation energy gradient of long-range corrected time-dependent density functional theory (LC-TDDFT) is presented. This is based on a previous analytical TDDFT gradient formalism, which avoids solving the coupled-perturbed Kohn-Sham equation for each nuclear degree of freedom. In LC-TDDFT, exchange interactions are evaluated by combining the short-range part of a DFT exchange functional with the long-range part of the Hartree-Fock exchange integral. This LC-TDDFT gradient was first examined by calculating the excited state geometries and adiabatic excitation energies of small typical molecules and a small protonated Schiff base. As a result, we found that long-range interactions play a significant role even in valence excited states of small systems. This analytical LC-TDDFT gradient was also applied to the investigations of small twisted intramolecular charge transfer (TICT) systems. By comparing with calculated ab initio multireference perturbation theory and experimental results, we found that LC-TDDFT gave much more accurate absorption and fluorescence energies of these systems than those of conventional TDDFTs using pure and hybrid functionals. For optimized excited state geometries, LC-TDDFT provided fairly different twisting and wagging angles of these small TICT systems in comparison with conventional TDDFT results.     

No-pair bonding in coinage metal dimers

High-level ab initio calculations at the coupled cluster with single and double substitutions and perturbative treatment of triple substitutions, CCSD(T), level of theory have been carried out for the dimers of coinage metal atoms Cu, Ag, and Au in the ground 1Sigma(g)+ state and in the excited 3Sigma(u)+ state. All of the calculations have been carried out with the inclusion of scalar-relativistic effects via the normalized elimination of the small component (NESC) method. For the dimers in the triplet state, nonzero bond dissociation energies are obtained which vary from 1.3 kcal/mol for 3Cu2 to 4.6 kcal/mol for 3Au2. Taking into account that, in bulky high-spin copper clusters, the bond dissociation energy per atom increases steeply to the value of ca. 19 kcal/mol, the results obtained in the present paper suggest that the bond dissociation energy per atom in high-spin gold clusters may reach extremely high values exceeding 20 kcal/mol thus becoming comparable to the usual bonding due to the spin-pairing mechanism.     

Analytical energy gradients for a unitary coupled-cluster theory

Our recently developed and tested unitary multiconfigurational coupled-cluster electronic wavefunction method is extended to permit, for the first time, the analytical evaluation of energy derivatives. The unitary nature of this method admits a variational energy functional whose stationary nature plays a key role in simplifying our derivation. Explicit expressions are given for the gradient (first energy derivative) for both the full unitary coupled cluster and its coupled electron pair approximation (CEPA).     

Analytical derivative techniques for TDDFT excited-state properties: Theory and application

We review our recent work on the methodology development of the excited-state properties for the molecules in vacuum and liquid solution.The general algorithms of analytical energy derivatives for the specific properties such as the first and second geometrical derivatives and IR/Raman intensities are demonstrated in the framework of the time-dependent density functional theory(TDDFT).The performance of the analytical approaches on the calculation of excited-state energy Hessian has also been shown.It is found that the analytical approaches are superior to the finite-difference method on the computational accuracy and efficiency.The computational cost for a TDDFT excited-state Hessian calculation is only 2–3 times as that for the DFT ground-state Hessian calculation.With the low computational complexity of the developed analytical approaches,it becomes feasible to realize the large-scale numerical calculations on the excited-state vibrational frequencies,vibrational spectroscopies and the electronic-structure parameters which enter the spectrum calculations of electronic absorption and emission,and resonance Raman spectroscopies for medium-to large-sized systems.     

On the accuracy of density-functional theory exchange-correlation functionals for H bonds in small water clusters: benchmarks approaching the complete basis set limit

The ability of several density-functional theory (DFT) exchange-correlation functionals to describe hydrogen bonds in small water clusters (dimer to pentamer) in their global minimum energy structures is evaluated with reference to second order Moller-Plesset perturbation theory (MP2). Errors from basis set incompleteness have been minimized in both the MP2 reference data and the DFT calculations, thus enabling a consistent systematic evaluation of the true performance of the tested functionals. Among all the functionals considered, the hybrid X3LYP and PBE0 functionals offer the best performance and among the nonhybrid generalized gradient approximation functionals, mPWLYP and PBE1W perform best. The popular BLYP and B3LYP functionals consistently underbind and PBE and PW91 display rather variable performance with cluster size.     

First principles calculations of electronic and optical properties of Ag2S

A theoretical study of structural, electronic and optical properties of Ag₂S is presented using the full potential linearized augmented plane wave (FP-LAPW) method within density functional theory (DFT). In this approach, the modified Becke Johnson (MBJ) potential coupled with Local Density Approximation (LDA) was used for the exchange-correlation potential calculation. Ground state properties are determined for the bulk material in monoclinic phase. Band structure reveals that this compound is a direct energy band gap semiconductor. MBJLDA results for the band gap of this compound are much better than those obtained using LDA, Perdew–Burke–Ernzerhof generalized gradient approximation (PBE-GGA) and Engel–Vosko's GGA (EV-GGA). A very good agreement is observed between MBJLDA band gap and corresponding experimental values as compared to other calculations. Optical constants including the dielectric function, refractive index, extinction coefficient, electron energy loss function, reflectivity and absorption coefficient are obtained and discussed.     

Analytic gradients for density cumulant functional theory: The DCFT-06 model

Density cumulant functional theory (DCFT) is one of a number of nascent electron correlation methods that are derived from reduced density matrices and cumulants thereof, instead of the wavefunction. Deriving properties from the density cumulant naturally yields methods that are size extensive and size consistent. In this work, we derive expressions for the analytic gradient, with respect to an external perturbation, for the DCFT-06 variant of density cumulant functional theory. Despite the fact that the DCFT-06 energy functional is stationary with respect to the density cumulant, the analytic gradients of the energy require the solution of perturbation-independent equations for both orbital and cumulant response. These two sets of linear response equations are coupled in nature and are solved iteratively with the solution of orbital and cumulant response equations each macroiteration, exhibiting rapid convergence. The gradients are implemented and benchmarked against coupled cluster theory with single and double excitations (CCSD) and CCSD with perturbative triple excitations [CCSD(T)], as well as accurate empirically corrected experimental data, for a test set comprising 15 small molecules. For most of the test cases, results from DCFT-06 are closer to CCSD(T) and empirical data than those from CCSD. Although the total energy and analytic gradient have the same asymptotic scaling, the present experience shows that the computational cost of the gradient is significantly lower.     

What are the spectroscopic properties of HFC‐32? Answers from DFT

Although coupled cluster theory coupled to large basis sets can reach impressive accuracies for thermochemical and spectroscopic properties, it is still limited to small/medium sized molecules. Density functional theory (DFT) represents the working option for systems composed of hundreds to thousands heavy atoms. In this context, investigations are required aimed at characterizing the performances of the different density functionals (DF). This work focuses on the study of DFT performances in the prediction of spectroscopic properties, with particular attention to the vibrational problem, by focusing on the CH₂F₂ molecule as a test case. An extensive and systematic investigation is performed on several DFT model chemistries by testing their predictions of molecular constants and vibrational frequencies and intensities against CCSD(T)/aug‐cc‐pCVQZ data. B3LYP, B3PW91, B97‐1, PBE0, TPSSh, M05, M05‐2X, and B2PLYP DFs are used in conjunction with a variety of basis sets. Anharmonic frequencies are derived from the VPT2 treatment of anharmonic‐ and hybrid CCSD(T)/DFT‐force fields. A software for VPT2 computations is also presented. © 2014 Wiley Periodicals, Inc.     

ER-BOC calculations of crystal bulk properties from smallest-cluster models

Ab initio calculation of bulk properties of crystals with a high accuracy, which is a long-time goal of solid chemistry and physics, is still difficult and expensive because a large cluster is required as a crystal structure model. This article proposes a model based on density functional theory (DFT) quantum chemistry calculations and the assumption that the bond order of a given atom with its nearest atoms in a compound is conserved over the entire range from its diatomic molecules to clusters and further to crystals. This entire range bond order conservation (ER-BOC) provides an effective way to correlate bulk properties of crystals with those of the corresponding molecules and small clusters. By combining this ER-BOC principle with hybrid DFT quantum chemistry calculations, accurate predictions of the bulk bond lengths of a crystal can be made using calculations on small clusters.     

Calculations of static and dynamic polarizabilities of excited states by means of density functional theory

We present density functional theory and calculations for excited state second order, static or dynamic, properties. The excited state properties are identified from a double residue of a cubic response function. The performance of various functionals, including the generalized gradient approximation and fractional exact Hartree-Fock exchange, is compared to coupled cluster calculations. Applications on excited state polarizabilities of s-tetrazine and pyrimidine show a good agreement with ab initio correlated, coupled cluster, results.