Response calculations based on an independent particle system with the exact one-particle density matrix: excitation energies

Adiabatic response time-dependent density functional theory (TDDFT) suffers from the restriction to basically an occupied → virtual single excitation formulation. Adiabatic time-dependent density matrix functional theory allows to break away from this restriction. Problematic excitations for TDDFT, viz. bonding-antibonding, double, charge transfer, and higher excitations, are calculated along the bond-dissociation coordinate of the prototype molecules H(2) and HeH(+) using the recently developed adiabatic linear response phase-including (PI) natural orbital theory (PINO). The possibility to systematically increase the scope of the calculation from excitations out of (strongly) occupied into weakly occupied ("virtual") natural orbitals to larger ranges of excitations is explored. The quality of the PINO response calculations is already much improved over TDDFT even when the severest restriction is made, to virtually the size of the TDDFT diagonalization problem (only single excitation out of occupied orbitals plus all diagonal doubles). Further marked improvement is obtained with moderate extension to allow for excitation out of the lumo and lumo+1, which become fractionally occupied in particular at longer distances due to left-right correlation effects. In the second place the interpretation of density matrix response calculations is elucidated. The one-particle reduced density matrix response for an excitation is related to the transition density matrix to the corresponding excited state. The interpretation of the transition density matrix in terms of the familiar excitation character (single excitations, double excitations of various types, etc.) is detailed. The adiabatic PINO theory is shown to successfully resolve the problematic cases of adiabatic TDDFT when it uses a proper PI orbital functional such as the PILS functional.     

Exact-exchange density functional theory for hyperpolarizabilities

Time-dependent density functional theory (TDDFT) employing the exact-exchange functional (TDDFTx) has been formulated using the optimized effective potential method for the beta static hyperpolarizabilities, where it reduces to coupled-perturbed Kohn-Sham theory. A diagrammatic technique is used to take the functional derivatives for the derivation of the adiabatic second kernel, which is required for the analytical calculation of the beta static hyperpolarizabilities with DFT. The derived formulas have been implemented using Gaussian basis sets. The structure of the adiabatic exact-exchange second kernel is described and numerical examples are presented. It is shown that no current DFT functional satisfies the correct properties of the second kernel. Not surprisingly, TDDFTx, which corrects the self-interaction error in standard DFT methods and has the correct long-range behavior, provides results close to those of time-dependent Hartree-Fock in the static limit.     

The adiabatic approximation in time-dependent density matrix functional theory: response properties from dynamics of phase-including natural orbitals

The adiabatic approximation is problematic in time-dependent density matrix functional theory. With pure density matrix functionals (invariant under phase change of the natural orbitals) it leads to lack of response in the occupation numbers, hence wrong frequency dependent responses, in particular α(ω→0)≠α(0) (the static polarizability). We propose to relinquish the requirement that the functional must be a pure one-body reduced density matrix (1RDM) functional, and to introduce additional variables which can be interpreted as phases of the one-particle states of the independent particle reference system formed with the natural orbitals, thus obtaining so-called phase-including natural orbital (PINO) functionals. We also stress the importance of the correct choice of the complex conjugation in the two-electron integrals in the commonly used functionals (they should not be of exchange type). We demonstrate with the Lo?wdin-Shull energy expression for two-electron systems, which is an example of a PINO functional, that for two-electron systems exact responses (polarizabilities, excitation energies) are obtained, while writing this energy expression in the usual way as a 1RDM functional yields erroneous responses.     

Excitation energies from range-separated time-dependent density and density matrix functional theory

Time-dependent density functional theory (TD-DFT) in the adiabatic formulation exhibits known failures when applied to predicting excitation energies. One of them is the lack of the doubly excited configurations. On the other hand, the time-dependent theory based on a one-electron reduced density matrix functional (time-dependent density matrix functional theory, TD-DMFT) has proven accurate in determining single and double excitations of H(2) molecule if the exact functional is employed in the adiabatic approximation. We propose a new approach for computing excited state energies that relies on functionals of electron density and one-electron reduced density matrix, where the latter is applied in the long-range region of electron-electron interactions. A similar approach has been recently successfully employed in predicting ground state potential energy curves of diatomic molecules even in the dissociation limit, where static correlation effects are dominating. In the paper, a time-dependent functional theory based on the range-separation of electronic interaction operator is rigorously formulated. To turn the approach into a practical scheme the adiabatic approximation is proposed for the short- and long-range components of the coupling matrix present in the linear response equations. In the end, the problem of finding excitation energies is turned into an eigenproblem for a symmetric matrix. Assignment of obtained excitations is discussed and it is shown how to identify double excitations from the analysis of approximate transition density matrix elements. The proposed method used with the short-range local density approximation (srLDA) and the long-range Buijse-Baerends density matrix functional (lrBB) is applied to H(2) molecule (at equilibrium geometry and in the dissociation limit) and to Be atom. The method accounts for double excitations in the investigated systems but, unfortunately, the accuracy of some of them is poor. The quality of the other excitations is in general much better than that offered by TD-DFT-LDA or TD-DMFT-BB approximations if the range-separation parameter is properly chosen. The latter remains an open problem.     

Generator coordinate method in time-dependent density-functional theory: memory made simple

The generator coordinate (GC) method is a variational approach to the quantum many-body problem in which interacting many-body wave functions are constructed as superpositions of (generally nonorthogonal) eigenstates of auxiliary Hamiltonians containing a deformation parameter. This paper presents a time-dependent extension of the GC method as a new approach to improve existing approximations of the exchange-correlation (XC) potential in time-dependent density-functional theory (TDDFT). The time-dependent GC method is shown to be a conceptually and computationally simple tool to build memory effects into any existing adiabatic XC potential. As an illustration, the method is applied to driven parametric oscillations of two interacting electrons in a harmonic potential (Hooke's atom). It is demonstrated that a proper choice of time-dependent generator coordinates in conjunction with the adiabatic local-density approximation reproduces the exact linear and nonlinear two-electron dynamics quite accurately, including features associated with double excitations that cannot be captured by TDDFT in the adiabatic approximation.     

Approximate molecular orbital theory: the ese MO formalism. I. General theory and notation

A new formalism for simplified molecular orbital calculations is elucidated. The focus of the formalism is the production of a good approximation to the LCAO SCF F matrix of Roothaan's equations, and preoccupation with approximations to individual molecular integrals is avoided. The great majority of multicentre two-electron integrals of the exact formalism are found to be largely inconsequential to the attainment of a good approximation to the F matrix.     

A density functional study of the electronic structure and spin Hamiltonian parameters of mononuclear thiomolybdenyl complexes

The electron paramagnetic resonance spin Hamiltonian parameters of mononuclear thiomolybdenyl complexes based upon the tris(pyrazolyl)borate ligand, together with their molybdenyl analogues, are calculated using density functional theory. The electronic g matrix and 95Mo hyperfine matrix are calculated as second-order response properties from the coupled-perturbed Kohn-Sham equations. The scalar relativistic zero-order regular approximation (ZORA) is used with an all-electron basis and an accurate mean-field spin-orbit operator which includes all one- and two-electron terms. The principal values and relative orientations of the g and A interaction matrices obtained from the experimental spectra in a previous EPR study are compared with those obtained from unrestricted Kohn-Sham calculations at the BP86 and B3LYP level, and the latter are found to be in good quantitative agreement. A quasi-restricted approach is used to analyze the influence of the various molecular orbitals on g and A. In all complexes the ground state magnetic orbital is dX2-Y2-based and the orientation of the A matrix is directly related to the orientation of this orbital. The largest single contribution to the orientation of the g matrix arises from the spin-orbit coupling of the dYZ-based lowest-unoccupied molecular orbital into the ground state. A number of smaller, cumulative charge-transfer contributions augment the d-d contributions. A comparison of the theoretical EPR parameters obtained using both crystallographic and gas-phase geometry-optimized structures of Tp*MoO(bdt) (Tp* = hydrotris(3,5-dimethylpyrazol-1-yl)borate, bdt = 1,2-benzenedithiolate) suggests a correspondence between the metal-dithiolate fold angle and the angle of noncoincidence between g and A.     

Coupled-perturbed density-matrix functional theory equations. Application to static polarizabilities

Starting from the variational equations for the natural occupation numbers and the recently proposed eigenequations for the natural spin-orbitals, we derive coupled-perturbed density-matrix equations that furnish a linear response of the one-electron reduced density matrix to a static perturbation when the total energy is a functional of the one-electron reduced density matrix. Cases when some occupation numbers achieve exactly 0 or 1 or when the total number of the particles in a system is not preserved are taken into consideration. The scheme is applied to computing static polarizabilities from two simple density-matrix functionals. The behavior of the functionals is erratic and they provide only little or no improvement over the coupled-perturbed Hartree-Fock results.     

Analytical Hessian of electronic excited states in time-dependent density functional theory with Tamm-Dancoff approximation

We present the analytical expression and computer implementation for the second-order energy derivatives of the electronic excited state with respect to the nuclear coordinates in the time-dependent density functional theory (TDDFT) with Gaussian atomic orbital basis sets. Here, the Tamm-Dancoff approximation to the full TDDFT is adopted, and therefore the formulation process of TDDFT excited-state Hessian is similar to that of configuration interaction singles (CIS) Hessian. However, due to the replacement of the Hartree-Fock exchange integrals in CIS with the exchange-correlation kernels in TDDFT, many quantitative changes in the derived equations are arisen. The replacement also causes additional technical difficulties associated with the calculation of a large number of multiple-order functional derivatives with respect to the density variables and the nuclear coordinates. Numerical tests on a set of test molecules are performed. The simulated excited-state vibrational frequencies by the analytical Hessian approach are compared with those computed by CIS and the finite-difference method. It is found that the analytical Hessian method is superior to the finite-difference method in terms of the computational accuracy and efficiency. The numerical differentiation can be difficult due to root flipping for excited states that are close in energy. TDDFT yields more exact excited-state vibrational frequencies than CIS, which usually overestimates the values.     

Coherent control and time-dependent density functional theory: towards creation of wave packets by ultrashort laser pulses

Explicitly time-dependent density functional theory (TDDFT) is a formally exact theory, which can treat very large systems. However, in practice it is used almost exclusively in the adiabatic approximation and with standard ground state functionals. Therefore, if combined with coherent control theory, it is not clear which control tasks can be achieved reliably, and how this depends on the functionals. In this paper, we continue earlier work in order to establish rules that answer these questions. Specifically, we look at the creation of wave packets by ultrashort laser pulses that contain several excited states. We find that (i) adiabatic TDDFT only works if the system is not driven too far from the ground state, (ii) the permanent dipole moments involved should not differ too much, and (iii) these results are independent of the functional used. Additionally, we find an artifact that produces fluence-dependent excitation energies.     

Open-shell reduced density matrix functional theory

Open-shell reduced density matrix functional theory is established by investigating the domain of the exact functional. For spin states that are the ground state, a particularly simple set is found to be the domain. It cannot be generalized to other spin states. A number of conditions satisfied by the exact density matrix functional is formulated and tested for approximate functionals. The exact functional does not suffer from fractional spin error, which is the source of the static correlation error in dissociated molecules. We prove that a simple approximation (called the Buijse-Baerends functional, Mu?ller or square root functional) has a non-positive fractional spin error. In the case of the H atom the error is zero. Numerical results for a few atoms are given for approximate density and density matrix functionals as well as a recently developed range-separated combination of both.     

Time-dependent density-functional theory beyond the adiabatic approximation: insights from a two-electron model system

Most applications of time-dependent density-functional theory (TDDFT) use the adiabatic local-density approximation (ALDA) for the dynamical exchange-correlation potential V(xc)(r,t). An exact (i.e., nonadiabatic) extension of the ground-state LDA into the dynamical regime leads to a V(xc)(r,t) with a memory, which causes the electron dynamics to become dissipative. To illustrate and explain this nonadiabatic behavior, this paper studies the dynamics of two interacting electrons on a two-dimensional quantum strip of finite size, comparing TDDFT within and beyond the ALDA with numerical solutions of the two-electron time-dependent Schrodinger equation. It is shown explicitly how dissipation arises through multiple particle-hole excitations, and how the nonadiabatic extension of the ALDA fails for finite systems but becomes correct in the thermodynamic limit.     

Approximate variational coupled cluster theory

We show that it is possible to construct an accurate approximation to the variational coupled cluster method, limited to double substitutions, from the minimization of a functional that is rigorously extensive, exact for isolated two-electron subsystems and invariant to transformations of the underlying orbital basis. This approximate variational coupled cluster theory is a modification and enhancement of our earlier linked pair functional theory. It is first motivated by the constraint that the inverse square root of the matrix that transforms the cluster amplitudes must exist. Low-order corrections are then included to enhance the accuracy of the approximation of variational coupled cluster, while ensuring that the computational complexity of the method never exceeds that of the standard traditional coupled cluster method. The effects of single excitations are included by energy minimization with respect to the orbitals defining the reference wavefunction. The resulting quantum chemical method is demonstrated to be a robust approach to the calculation of molecular electronic structure and performs well when static correlation effects are strong.     

Time-dependent density-functional and reduced density-matrix methods for few electrons: Exact versus adiabatic approximations

To address the impact of electron correlations in the linear and non-linear response regimes of interacting many-electron systems exposed to time-dependent external fields, we study one-dimensional (1D) systems where the interacting problem is solved exactly by exploiting the mapping of the 1D N-electron problem onto an N-dimensional single electron problem. We analyze the performance of the recently derived 1D local density approximation as well as the exact-exchange orbital functional for those systems. We show that the interaction with an external resonant laser field shows Rabi oscillations which are detuned due to the lack of memory in adiabatic approximations. To investigate situations where static correlations play a role, we consider the time-evolution of the natural occupation numbers associated to the reduced one-body density matrix. Those studies shed light on the non-locality and time-dependence of the exchange and correlation functionals in time-dependent density and density-matrix functional theories.     

Troubleshooting time-dependent density-functional theory for photochemical applications: oxirane

The development of analytic-gradient methodology for excited states within conventional time-dependent density-functional theory (TDDFT) would seem to offer a relatively inexpensive alternative to better established quantum-chemical approaches for the modeling of photochemical reactions. However, even though TDDFT is formally exact, practical calculations involve the use of approximate functional, in particular the TDDFT adiabatic approximation, the use of which in photochemical applications must be further validated. Here, we investigate the prototypical case of the symmetric CC ring opening of oxirane. We demonstrate by direct comparison with the results of high-quality quantum Monte Carlo calculations that, far from being an approximation on TDDFT, the Tamm-Dancoff approximation is a practical necessity for avoiding triplet instabilities and singlet near instabilities, thus helping maintain energetically reasonable excited-state potential energy surfaces during bond breaking. Other difficulties one would encounter in modeling oxirane photodynamics are pointed out.     

Nonadiabatic couplings from time-dependent density functional theory: formulation in the Casida formalism and practical scheme within modified linear response

We present an efficient method to compute nonadiabatic couplings (NACs) between the electronically ground and excited states of molecules, within the framework of time-dependent density functional theory (TDDFT) in frequency domain. Based on the comparison of dynamic polarizability formulated both in the many-body wave function form and the Casida formalism, a rigorous expression is established for NACs, which is similar to the calculation of oscillator strength in the Casida formalism. The adiabatic local density approximation (ALDA) gives results in reasonable accuracy as long as the conical intersection (ci) is not approached too closely, while its performance quickly degrades near the ci point. This behavior is consistent with the real-time TDDFT calculation. Through the use of modified linear response theory together with the ground-state-component separation scheme, the performance of ALDA can be greatly improved, not only in the vicinity of ci but also for Rydberg transitions and charge-transfer excitations. Several calculation examples, including the quantization of NACs from the Jahn-Teller effect in the H3 system, have been given to show that TDDFT can efficiently give NACs with an accuracy comparable to that of wave-function-based methods.     

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.     

Density matrix approach to orbital relaxation dynamics in ionization

Dynamical response of electrons to a hole generated during ionization is formulated in time domain with the density matrix equations in the time‐dependent unrestricted Hartree–Fock approximation. Time evolutions of orbital energies and electron‐density distributions are computed for K‐shell and M‐shell ionizations of a Na atom by taking into account nonlinear coupling of density matrices beyond linear response. When the hole is generated so slowly that the adiabatic theorem is satisfied, the simulation eventually converges to the state of a fully relaxed Na⁺ ion. A rapid generation of a K‐shell hole (within about 1 fs) leads to a breakdown of the adiabatic theorem, triggering a collective oscillation of the electrons with the period of sub‐femtoseconds. The shake‐up effect associated with strong orbital relaxation in inner‐shell ionization is manifested as a mixing of occupied and unoccupied states in the density matrix.     

Intermolecular dispersion energies from time-dependent density functional theory

Non-expanded dispersion energies are calculated from time-dependent coupled-perturbed density functional theory (DFT) employing various non-hybrid and hybrid exchange-correlation potentials and suitable adiabatic local density approximations for the exchange-correlation kernel. Considering the dimer systems He₂, Ne₂, Ar₂, NeAr, NeHF, ArHF, (H₂)₂, (HF)₂, and (H₂O)₂ it is shown that the effects of intramonomer electron correlation on the dispersion energy are accurately reproduced with the PBE0AC exchange-correlation potential. In contrast, the uncoupled sum-over-states approximation yields inacceptable errors. These are mainly due to neglect of the Coulomb and exchange-correlation kernels and therefore, not substantially improved through an asymptotic correction of the exchange-correlation potential.