Linear-scaling quantum mechanical methods for excited states

The poor scaling of many existing quantum mechanical methods with respect to the system size hinders their applications to large systems. In this tutorial review, we focus on latest research on linear-scaling or O(N) quantum mechanical methods for excited states. Based on the locality of quantum mechanical systems, O(N) quantum mechanical methods for excited states are comprised of two categories, the time-domain and frequency-domain methods. The former solves the dynamics of the electronic systems in real time while the latter involves direct evaluation of electronic response in the frequency-domain. The localized density matrix (LDM) method is the first and most mature linear-scaling quantum mechanical method for excited states. It has been implemented in time- and frequency-domains. The O(N) time-domain methods also include the approach that solves the time-dependent Kohn-Sham (TDKS) equation using the non-orthogonal localized molecular orbitals (NOLMOs). Besides the frequency-domain LDM method, other O(N) frequency-domain methods have been proposed and implemented at the first-principles level. Except one-dimensional or quasi-one-dimensional systems, the O(N) frequency-domain methods are often not applicable to resonant responses because of the convergence problem. For linear response, the most efficient O(N) first-principles method is found to be the LDM method with Chebyshev expansion for time integration. For off-resonant response (including nonlinear properties) at a specific frequency, the frequency-domain methods with iterative solvers are quite efficient and thus practical. For nonlinear response, both on-resonance and off-resonance, the time-domain methods can be used, however, as the time-domain first-principles methods are quite expensive, time-domain O(N) semi-empirical methods are often the practical choice. Compared to the O(N) frequency-domain methods, the O(N) time-domain methods for excited states are much more mature and numerically stable, and have been applied widely to investigate the dynamics of complex molecular systems.     

Simulation of Nanomaterials

In this talk Ⅰ will discuss three developments required for simulating nano-scale devices, namely, (1) linear-scaling calculation for excited states, (2) beyond the Born-Oppenheimer approximation, and (3) quantum chemistry simulation of open systems. Linear-scaling localized-density-matrix (LDM) method has been developed for calculating accurately the excited state properties, for instance, absorption spectrum. It has been implemented with the PPP, CNDO/S, INDO/S, AM1 and PM3 semiempirical Hamiltonians, and applied to nano-sized PPV aggregates and carbon nanotubes. Nuclear dynamics has been simulated simultaneously with the electronic dynamics in the time domain, which is within the framework of the LDM method. Exchange of energy and electrons with the surrounding has been included as well, and interesting results on excited state relaxation and corresponding optical responses will be presented.     

Linear-scaling computation of excited states in time-domain

The applicability of quantum mechanical methods is severely limited by their poor scaling.To circumvent the problem,linearscaling methods for quantum mechanical calculations had been developed.The physical basis of linear-scaling methods is the locality in quantum mechanics where the properties or observables of a system are weakly influenced by factors spatially far apart.Besides the substantial efforts spent on devising linear-scaling methods for ground state,there is also a growing interest in the development of linear-scaling methods for excited states.This review gives an overview of linear-scaling approaches for excited states solved in real time-domain.     

Theoretical study of low-lying excited states of molecular aggregates. I. Development of linear-scaling TD-DFT

The project aims to develop an integrated linear-scaling time-dependent density functional theory (TD-DFT) for studying low-lying excited states of luminescent molecular materials, especially those fluorescence and phosphorescence co-emitting systems. The central idea will be "from fragments to molecule" (FF2M). That is, the fragmental information will be employed to synthesize the molecular wave function, such that the locality (transferability) of the fragments (functional groups) is directly built into the algorithms. Both relativistic and spin-adapted open-shell TD-DFT will be considered. Use of the renormalized exciton method will also be made to further enhance the efficiency and accuracy of TD-DFT. Solvent effects are to be targeted with the fragment-based solvent model. It is expected that the integrated TD-DFT and program will be of great value in rational design of luminescent molecular materials.     

Linear-scaling implementation of molecular electronic self-consistent field theory

A linear-scaling implementation of Hartree-Fock and Kohn-Sham self-consistent field (SCF) theories is presented and illustrated with applications to molecules consisting of more than 1000 atoms. The diagonalization bottleneck of traditional SCF methods is avoided by carrying out a minimization of the Roothaan-Hall (RH) energy function and solving the Newton equations using the preconditioned conjugate-gradient (PCG) method. For rapid PCG convergence, the Lowdin orthogonal atomic orbital basis is used. The resulting linear-scaling trust-region Roothaan-Hall (LS-TRRH) method works by the introduction of a level-shift parameter in the RH Newton equations. A great advantage of the LS-TRRH method is that the optimal level shift can be determined at no extra cost, ensuring fast and robust convergence of both the SCF iterations and the level-shifted Newton equations. For density averaging, the authors use the trust-region density-subspace minimization (TRDSM) method, which, unlike the traditional direct inversion in the iterative subspace (DIIS) scheme, is firmly based on the principle of energy minimization. When combined with a linear-scaling evaluation of the Fock/Kohn-Sham matrix (including a boxed fitting of the electron density), LS-TRRH and TRDSM methods constitute the linear-scaling trust-region SCF (LS-TRSCF) method. The LS-TRSCF method compares favorably with the traditional SCF/DIIS scheme, converging smoothly and reliably in cases where the latter method fails. In one case where the LS-TRSCF method converges smoothly to a minimum, the SCF/DIIS method converges to a saddle point.     

Time-dependent density-functional theory/localized density matrix method for dynamic hyperpolarizability

Time-dependent density-functional theory/localized density matrix method (TDDFT/LDM) was developed to calculate the excited state energy, absorption spectrum and dynamic polarizability. In the present work we generalize it to calculate the dynamic hyperpolarizabilities in both time and frequency domains. We show that in the frequency domain the 2n+1 rule can be derived readily and the dynamic hyperpolarizabilities are thus calculated efficiently. Although the time-domain TDDFT/LDM is time consuming, its implementation is straightforward because the evaluation of the derivatives of exchange-correlation potential with respect to electron density is avoided. Moreover, the time-domain method can be used to simulate higher order response which is very difficult to be calculated with the frequency-domain method.     

Effective local potentials for excited states

The constrained variational Hartree-Fock method for excited states of the same symmetry as the ground state [Chem. Phys. Lett. 287, 189 (1998)] is combined with the effective local potential (ELP) method [J. Chem. Phys. 125, 081104 (2006)] to generate Kohn-Sham-type exact-exchange potentials for singly excited states of many-electron systems. Illustrative examples include the three lowest (2)S states of the Li and Na atoms and the three lowest (3)S states of He and Be. For the systems studied, excited-state ELPs differ from the corresponding ground-state potentials in two respects: They are less negative and have small additional "bumps" in the outer electron region. The technique is general and can be used to approximate excited-state exchange-correlation potentials for other orbital-dependent functionals.     

How does it become possible to treat delocalized and/or open-shell systems in fragmentation-based linear-scaling electronic structure calculations? The case of the divide-and-conquer method

The authors have developed a fragmentation-based linear-scaling electronic structure calculation strategy named the divide-and-conquer (DC) method, which has been implemented into the Gamess program package. Although there are many sorts of fragmentation-based linear-scaling schemes, most of them require the charge and spin multiplicity of each fragment a priori. Therefore, their applications to delocalized and/or open-shell systems have been limited. However, the DC method is a notable exception because the distribution of electrons in the entire system is automatically determined by the universal Fermi level. In this perspective, the authors have summarized the performance of the linear-scaling self-consistent field (SCF) and post-SCF calculations of delocalized and/or open-shell systems based on the DC method. Furthermore, some future prospects of the method have been discussed.     

Vibronic interactions and possible electron pairing in the photoinduced excited electronic States in molecular systems: a theoretical study

Electron-phonon interactions in the photoinduced excited electronic states in molecular systems such as phenanthrene-edge-type hydrocarbons are discussed and compared with those in the monoanions and cations. The complete phase patterns difference between the highest occupied molecular orbitals (HOMO) and the lowest unoccupied molecular orbitals (LUMO) (the atomic orbitals between two neighboring carbon atoms combined in phase (out of phase) in the HOMO are combined out of phase (in phase) in the LUMO) are the main reason that the C-C stretching modes around 1500 cm(-1) afford much larger electron-phonon coupling constants in the excited electronic states than in the charged electronic states. The frequencies of the vibrational modes that play an essential role in the electron-phonon interactions for the excited electronic states are similar to those for the monoanions and cations in phenanthrene-edge-type hydrocarbons. Possible electron pairing and Bose-Einstein condensation in the photoinduced excited electronic states as well as those in the monoanions and cations in molecular systems such as phenanthrene-edge-type hydrocarbons are also discussed.     

Linear-scaling multipole-accelerated Gaussian and finite-element Coulomb method

A linear-scaling implementation of the Gaussian and finite-element Coulomb (GFC) method is presented for the rapid computation of the electronic Coulomb potential. The current work utilizes the fast multipole method (FMM) for the evaluation of the Poisson equation boundary condition. The FMM affords significant savings for small- and medium-sized systems and overcomes the bottleneck in the GFC method for very large systems. Compared to an exact analytical treatment of the boundary, more than 100-fold speedups are observed for systems with more than 1000 basis functions without any significant loss of accuracy. We present CPU times to demonstrate the effectiveness of the linear-scaling GFC method for both one-dimensional polyalanine chains and the challenging case of three-dimensional diamond fragments.     

Variable metric optimization of molecular geometry in electronically excited states

The variable metric (VM) method is used to optimize molecular geometry in electronically excited states. A general expression for the first derivative of energy in the particular excited state is derived, considering configuration interaction of all singly excited configurations. A special expression for the excited states energy derivative is given for calculations with semiempirical methods of CNDO type. The geometry optimizations of a set of molecules in various excited states have been carried out by the CNDO/2 method. The results of computations have been discussed and compared with the available experimental data. A good agreement of the calculated geometries with the experimental ones has been shown in the first excited states and a relatively good agreement in the higher states, with some exceptions. Some special features of the proposed method are discussed.     

Wave packet dynamics in triplet states of Na2 attached to helium nanodroplets

The dynamics of vibrational wave packets excited in Na2 dimers in the triplet ground and excited states is investigated by means of helium nanodroplet isolation (HENDI) combined with femtosecond pump-probe spectroscopy. Different pathways in the employed resonant multiphoton ionization scheme are identified. Within the precision of the method, the wave packet dynamics appears to be unperturbed by the helium droplet environment.     

Multi-configuration electron-hole potential method for excited states

The recently proposed electron-hole potential (EHP) method for excited states is extended to the multi-configurational case. The variation equation is solved using the quadratic convergence method. The EHP methods are shown to be approximations to the complete singly excited configuration interaction (CSECI) in the variational sense. Extended Brillouin theorems are proved for the EHP methods. The excitation energies and wave functions obtained by one and two configurational EHP methods agree well with those of the CSECI method. The EHP methods have clear advantage in the computer time requirement over the CI method and are especially suited for a calculation of approximate excited states of large molecules. The EHP methods are applicable to excited states which belong to the same irreducible representation as the ground state.     

Color tuning in photofunctional proteins

Depending on protein environment, a single photofunctional chromophore shows a wide variation of photoabsorption/emission energies. This photobiological phenomenon, known as color tuning, is observed in human visual cone pigments, firefly luciferase, and red fluorescent protein. We investigate the origin of color tuning by quantum chemical calculations on the excited states: symmetry-adapted cluster-configuration interaction (SAC-CI) method for excited states and a combined quantum mechanical (QM)/molecular mechanical (MM) method for protein environments. This Minireview summarizes our theoretical studies on the above three systems and explains a common feature of their color-tuning mechanisms. It also discuss the possibility of artificial color tuning toward a rational design of photoabsorption/emission properties.     

A density matrix-based method for the linear-scaling calculation of dynamic second- and third-order properties at the Hartree-Fock and Kohn-Sham density functional theory levels

A density matrix-based time-dependent self-consistent field (D-TDSCF) method for the calculation of dynamic polarizabilities and first hyperpolarizabilities using the Hartree-Fock and Kohn-Sham density functional theory approaches is presented. The D-TDSCF method allows us to reduce the asymptotic scaling behavior of the computational effort from cubic to linear for systems with a nonvanishing band gap. The linear scaling is achieved by combining a density matrix-based reformulation of the TDSCF equations with linear-scaling schemes for the formation of Fock- or Kohn-Sham-type matrices. In our reformulation only potentially linear-scaling matrices enter the formulation and efficient sparse algebra routines can be employed. Furthermore, the corresponding formulas for the first hyperpolarizabilities are given in terms of zeroth- and first-order one-particle reduced density matrices according to Wigner's (2n+1) rule. The scaling behavior of our method is illustrated for first exemplary calculations with systems of up to 1011 atoms and 8899 basis functions.     

Ab initio nonadiabatic dynamics study of ultrafast radiationless decay over conical intersections illustrated on the Na3F cluster

We present a theoretical approach for the ultrafast nonadiabatic dynamics based on the ab initio molecular dynamics carried out "on the fly" in the framework of the configuration interaction method combined with Tully's surface hopping algorithm for nonadiabatic transitions. This approach combined with our Wigner distribution approach allows us to perform accurate simulations of femtosecond pump-probe spectra in the systems where radiationless transitions among electronic states take place. In this paper we illustrate this by theoretical simulation of ultrafast processes and nonradiative relaxation in the Na(3)F cluster, involving three excited states and the ground electronic state. Furthermore, we show that our accurate simulation of the photoionization pump-probe spectrum is in full agreement with the experimental signal. Based on the nonadiabatic dynamics at high level of accuracy and taking into account all degrees of freedom, the nonradiative lifetime for the 1 (1)B(1) excited state of Na(3)F has been determined to be approximately 900 fs.     

Optimized effective potential method for individual low-lying excited states

This paper presents an optimized effective potential (OEP) approach based on density functional theory (DFT) for individual excited states that implements a simple method of taking the necessary orthogonality constraints into account. The amended Kohn-Sham (KS) equations for orbitals of excited states having the same symmetry as the ground one are proposed. Using a variational principle with some orthogonality constraints, the OEP equations determining a local exchange potential for excited states are derived. Specifically, local potentials are derived whose KS determinants minimize the total energies and are simultaneously orthogonal to the determinants for states of lower energies. The parametrized form of an effective DFT potential expressed as a direct mapping of the external potential is used to simplify the OEP integral equations. A performance of the presented method is examined by exchange-only calculations of excited state energies for simple atoms and molecules.     

Deflation techniques in quantum chemistry: Excited states from ground states

Applications of deflation techniques to the study of excited states of quantum systems are analyzed. It is demonstrated how these methods allow us to transform the excited state problem of one Hamiltonian, into the ground state problem of an auxiliary one. As an example, potential application in the density functional treatment of excited states is discussed. The inclusion of approximations in this scheme, such as the solution of the proposed model within a finite basis set is discussed. An extension of the Hartree–Fock (HF) method to excited states is presented. This new treatment includes previous self consistent field extensions to excited states and provides us with a way to obtain the HF extension to excited states of any ground state method. These results make the excited states of a system accessible through all ground state theoretical techniques. © 2013 Wiley Periodicals, Inc.