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.     

Linear-scaling localized-density-matrix method for the ground and excited states of one-dimensional molecular systems

A linear-scaling localized-density-matrix (LDM) method is developed to evaluate the ground-state reduced single-electron density matrices of one-dimensional molecular systems. The new method may be combined with the existing linear-scaling LDM method for the excited states (Yokojima and Chen, Chem. Phys. Lett. 292 (1998) 379), and thus leads to a linear-scaling calculation method for the properties of both the ground and excited states. The combined method is applied to the polyacetylene oligomers and the linear-scaling of the total computational time is clearly demonstrated.     

Achieving plane wave accuracy in linear-scaling density functional theory applied to periodic systems: a case study on crystalline silicon

Linear-scaling methods for density functional theory promise to revolutionize the scope and scale of first-principles quantum mechanical calculations. Crystalline silicon has been the system of choice for exploratory tests of such methods in the literature, yet attempts at quantitative comparisons under linear-scaling conditions with traditional methods or experimental results have not been forthcoming. A detailed study using the ONETEP code is reported here, demonstrating for the first time that plane wave accuracy can be achieved in linear-scaling calculations on periodic systems.     

GridMol2.0: Implementation and application of linear-scale quantum mechanics methods and molecular visualization

GridMol is a “one-stop” platform for molecular structure building, scientific computing, and molecular visualization aided by a high-performance computing environment. GridMol version 2.0 introduces two unique features: the first is fragment-based linear-scaling quantum chemistry methods, such as molecular fractionation with conjugate caps and fragment molecular orbital methods; the second is that GridMol enables users to visualize molecular geometries along a geometry optimization and an intrinsic reaction coordinate calculation. Compared with version 1.0, fragment-based linear-scaling quantum chemistry methods implemented in GridMol version 2.0 can be used as a useful tool for performing quantum calculations for large molecular systems to explore the mechanisms involved in protein-ligand or targeted drug interactions.     

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.     

Excited state properties of sizable molecules in solution: from structure to reactivity

We review some recent advances in quantum mechanical methods devised specifically for the study of excited electronic state of large size molecules in solution. The adopted theoretical/computational framework is rooted in the density functional theory (DFT) and its time-dependent extension (TD-DFT) for the characterization of ground and excited states, in the polarizable continuum model (PCM) for the treatment of bulk solvent effects, and in time-dependent quantum mechanical methods for chemical dynamics. Selected applications to the simulation of absorption spectra, to the interpretation of time-resolved experiments, and to the computation of dissociative electron transfer rates are presented and discussed.     

New method for direct linear-scaling calculation of electron density of proteins

A new scheme for direct linear-scaling quantum mechanical calculation of electron density of protein systems is developed. The new scheme gives much improved accuracy of electron density for proteins than the original MFCC (molecular fractionation with conjugate caps) approach in efficient linear-scaling calculation for protein systems. In this new approach, the error associated with each cut in the MFCC approach is estimated by computing the two neighboring amino acids in both cut and uncut calculations and is corrected. Numerical tests are performed on six oligopeptide taken from PDB (protein data bank), and the results show that the new scheme is efficient and accurate.     

Quantum chemical methods for the investigation of photoinitiated processes in biological systems: theory and applications.

With the advent of modern computers and advances in the development of efficient quantum chemical computer codes, the meaningful computation of large molecular systems at a quantum mechanical level became feasible. Recent experimental effort to understand photoinitiated processes in biological systems, for instance photosynthesis or vision, at a molecular level also triggered theoretical investigations in this field. In this Minireview, standard quantum chemical methods are presented that are applicable and recently used for the calculation of excited states of photoinitiated processes in biological molecular systems. These methods comprise configuration interaction singles, the complete active space self-consistent field method, and time-dependent density functional theory and its variants. Semiempirical approaches are also covered. Their basic theoretical concepts and mathematical equations are briefly outlined, and their properties and limitations are discussed. Recent successful applications of the methods to photoinitiated processes in biological systems are described and theoretical tools for the analysis of excited states are presented.     

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 combined quantum mechanical and molecular mechanical method using modified generalized hybrid orbitals: implementation for electronic excited states

The generalized hybrid orbital (GHO) method is implemented at the second-order approximate coupled cluster singles and doubles (CC2) level for quantum mechanical (QM)/molecular mechanical (MM) electronic excited state calculations. The linear response function of CC2 in the GHO scheme is derived and implemented. The new implementation is applied to the first singlet excited states of three aromatic amino acids, phenylalanine, tyrosine, and tryptophan, and also bacteriorhodopsin for assessment. The results obtained for aromatic amino acids agreed well with the full QM CC2 calculations, while the calculated excitation energies of bacteriorhodopsin and its chromophore, all-trans retinal, reproduced the environmental shift of the experimental data. For the bacteriorhodopsin case, the environmental shift of GHO also showed good agreements with the experimental data. The contribution of the quantum effect of certain moieties in the excited states is elucidated by changing the partitioning of QM and MM regions.     

Ultrafast coherent dynamics of nonadiabatically coupled quasi-degenerate excited states in molecules: Population and vibrational coherence transfers

Results of a theoretical study of ultrafast coherent dynamics of nonadiabatically coupled quasi-degenerate π-electronic excited states of molecules were presented. Analytical expressions for temporal behaviors of population and vibrational coherence were derived using a simplified model to clarify the quantum mechanical interferences between the two coherently excited electronic states, which appeared in the nuclear wavepacket simulations [M. Kanno, H. Kono, Y. Fujimura, S.H. Lin, Phys. Rev. Lett 104 (2010) 108302]. The photon-polarization direction of the linearly polarized laser, which controls the populations of the two quasi-degenerate electronic states, determines constructive or destructive interference. Features of the vibrational coherence transfer between the two coupled quasi-electronic states through nonadiabatic couplings are also presented. Information on both the transition frequency and nonadiabatic coupling matrix element between the two states can be obtained by analyzing signals of two kinds of quantum beats before and after transfer through nonadiabatic coupling.     

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.     

Rayleigh–ritz method for excited quantum states via nonlinear variations without constraints: Role of supersymmetry

Quantum mechanical variation principle in the form of energy minimization is applicable only to ground states of systems, or, at best, states of lowest energies of given symmetries, provided the symmetry information is embedded in chosen trial functions. Thus, for bound quantum states with specified choices of trial functions involving nonlinear parameters, scope of the principle is severely restricted. A pedagogic way out is to enforce exact orthogonality of the chosen function with all exact lower energy states. In actual practice, this limits one to opt for linear variations where upper bound to each state is obtained in a single run. In this work, the motivation is to explore if there exists at all a way to determine optimized wave functions and energies for excited states via nonlinear variations but without any constraints, even for simple systems. Realizing that the major problem in excited‐state nonlinear variations is concerned with the variations of nodal positions, at least for problems reducible to one dimension, we seek a route via which nodes could be fixed beforehand, so that the information gained may be subsequently utilized to construct a suitable nonlinear trial function and carry out a straightforward optimization. To achieve this end, the idea of supersymmetric quantum mechanics has been used quite profitably, yielding the nodal structure of the excited states. Workability of the strategy for several excited‐state wave functions and their properties is demonstrated by choosing the problems of spherical Stark effect on hydrogen atom and anharmonic oscillator. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Ab initio molecular dynamics and time-resolved photoelectron spectroscopy of electronically excited uracil and thymine

The reaction dynamics of excited electronic states in nucleic acid bases is a key process in DNA photodamage. Recent ultrafast spectroscopy experiments have shown multicomponent decays of excited uracil and thymine, tentatively assigned to nonadiabatic transitions involving multiple electronic states. Using both quantum chemistry and first principles quantum molecular dynamics methods we show that a true minimum on the bright S2 electronic state is responsible for the first step that occurs on a femtosecond time scale. Thus the observed femtosecond decay does not correspond to surface crossing as previously thought. We suggest that subsequent barrier crossing to the minimal energy S2/S1 conical intersection is responsible for the picosecond decay.     

Calculation of the excited states properties of LH1 complex of Thermochromatium tepidum

Calculation of the excited states properties of pigment complexes is one of the key problems in the photosynthesis research. The excited states of LH1 complex of Thermochromatium tepidum were studied by means of the high‐precision quantum chemistry methods. The influence of different parameters of the calculation procedure was examined. The optimal scheme of calculation was chosen by comparison of calculated results with the experimental data on absorption, electronic and magnetic circular dichroism spectra. The high importance of the account of the second excited states of bacteriochlorophylls and of site heterogeneity was shown. © 2018 Wiley Periodicals, Inc.