Beyond Time-Dependent Density Functional Theory Using Only Single Excitations: Methods for Computational Studies of Excited States in Complex Systems

Single-excitation methods, namely, configuration interaction singles and time-dependent density functional theory (TDDFT), along with semiempirical versions thereof, represent the most computationally affordable electronic structure methods for describing electronically excited states, scaling as [Formula: see text] absent further approximations. This relatively low cost, combined with a treatment of electron correlation, has made TDDFT the most widely used excited-state quantum chemistry method over the past 20+ years. Nevertheless, certain inherent problems (beyond just the accuracy of this or that exchange-correlation functional) limit the utility of traditional TDDFT. For one, it affords potential energy surfaces whose topology is incorrect in the vicinity of any conical intersection (CI) that involves the ground state. Since CIs are the conduits for transitions between electronic states, the TDDFT description of photochemistry (internal conversion and intersystem crossing) is therefore suspect. Second, the [Formula: see text] cost can become prohibitive in large systems, especially those that involve multiple electronically coupled chromophores, for example, the antennae structures of light-harvesting complexes or the conjugated polymers used in organic photovoltaics. In such cases, the smallest realistic mimics might already be quite large from the standpoint of ab initio quantum chemistry. This Account describes several new computational methods that address these problems. Topology around a CI can be rigorously corrected using a "spin-flip" version of TDDFT, which involves an α → β spin-flipping transition in addition to occupied → virtual excitation of one electron. Within this formalism, singlet states are generated via excitation from a high-spin triplet reference state, doublets from a quartet, etc. This provides a more balanced treatment of electron correlation between ground and excited states. Spin contamination is problematic away from the Franck-Condon region, but we describe a "spin-complete" version of the theory in which proper spin eigenstates are obtained by construction. For systems of coupled chromophores, we have developed an ab initio version of the Frenkel-Davydov exciton model in which collective excitations of the system are expanded in a basis of excited states computed for individual chromophores. The monomer calculations are trivially parallelizable, as is computation of the coupling matrix elements needed to construct the exciton Hamiltonian, and systems containing hundreds of chromophores can be tackled on commodity hardware. This enables calculations on organic semiconductors, where even small model systems exhibit a semicontinuum of excited states that renders traditional TDDFT computationally challenging. Despite including only single excitations on each monomer, the exciton model can describe entangled spins on two or more monomers, an effect that is responsible for excitation energy transfer between chromophores, for example, in singlet fission. Excitonic approximations can also be applied to the TDDFT equations themselves, and a particularly promising application is to describe the effects of environment on an excitation that is localized on a single chromophore. This "local excitation approximation" to TDDFT allows an essentially arbitrary number of solvent molecules to be included in the calculation in a highly parallelizable way such that the time-to-solution increases only very slowly as additional solvent molecules are added. It is therefore possible to converge the calculation with respect to describing an ever-larger portion of the environment at a quantum-mechanical level.

Control of Nonadiabatic Passage through a Conical Intersection by a Dynamic Resonance

Nonadiabatic processes, dominated by dynamic passage of reactive fluxes through conical intersections (CIs), are considered to be appealing means for manipulating reaction paths, particularly via initial vibrational preparation. Nevertheless, obtaining direct experimental evidence of whether specific-mode excitation affects the passage at the CI is challenging, requiring well-resolved time- or frequency-domain experiments. Here promotion of methylamine-d2 (CH3ND2) molecules to spectral-resolved rovibronic states on the excited S1 potential energy surface, coupled to sensitive D photofragment probing, allowed us to follow the N-D bond fission dynamics. The branching ratios between slow and fast D photofragments and the internal energies of the CH3ND(X̃) photofragments confirm correlated anomalies for predissociation initiated from specific rovibronic states. These anomalies reflect the existence of a dynamic resonance that strongly depends on the energy of the initially excited rovibronic states, the evolving vibrational mode on the repulsive S1 part during N-D bond elongation, and the manipulated passage through the CI that leads to CH3ND radicals excited with C-N-D bending. This resonance plays an important role in the bifurcation dynamics at the CI and can be foreseen to exist in other photoinitiated processes and to control their outcome.

Gas-Phase Femtosecond Particle Spectroscopy: A Bottom-Up Approach to Nucleotide Dynamics

We summarize how gas-phase ultrafast charged-particle spectroscopy has been used to provide an understanding of the photophysics of DNA building blocks. We focus on adenine and discuss how, following UV excitation, specific interactions determine the fates of its excited states. The dynamics can be probed using a systematic bottom-up approach that provides control over these interactions and that allows ever-larger complexes to be studied. Starting from a chromophore in adenine, the excited state decay mechanisms of adenine and chemically substituted or clustered adenine are considered and then extended to adenosine mono-, di-, and trinucleotides. We show that the gas-phase approach can offer exquisite insight into the dynamics observed in aqueous solution, but we also highlight stark differences. An outlook is provided that discusses some of the most promising developments in this bottom-up approach.

Quantum Mechanical Studies on the Photophysics and the Photochemistry of Nucleic Acids and Nucleobases

The photophysics and photochemistry of DNA is of great importance due to the potential damage of the genetic code by UV light. Quantum mechanical studies have played a key role in interpretating the results of modern time-resolved pump-probe spectroscopy, and in elucidating the main photoactivated reactive paths. This review provides a concise, complete picture of the computational studies carried out, approximately, in the past decade. We start with an overview of the photophysics of the nucleobases in the gas phase and in solution. We discuss the proposed mechanisms for ultrafast decay to the ground state, that involve conical intersections, consider the role of triplet states, and analyze how the solvent modulates the photophysics. Then we move to larger systems, from dinucleotides to single- and double-stranded oligonucleotides. We focus on the possible role of charge transfer and delocalized or excitonic states in the photophysics of these systems and discuss the main photochemical paths. We finish with an outlook on the current challenges in the field and future directions of research.

Experimental Detection of Branching at a Conical Intersection in a Highly Fluorescent Molecule

Conical intersections are molecular configurations at which adiabatic potential-energy surfaces touch. They are predicted to be ubiquitous, yet condensed-phase experiments have focused on the few systems with clear spectroscopic signatures of negligible fluorescence, high photoactivity, or femtosecond electronic kinetics. Although rare, these signatures have become diagnostic for conical intersections. Here we detect a coherent surface-crossing event nearly two picoseconds after optical excitation in a highly fluorescent molecule that has no photoactivity and nanosecond electronic kinetics. Time-frequency analysis of high-sensitivity measurements acquired using sub-8 fs pulses reveals phase shifts of the signal due to branching of the wavepacket through a conical intersection. The time-frequency analysis methodology demonstrated here on a model compound will enable studies of conical intersections in molecules that do not exhibit their diagnostic signatures. Improving the ability to detect conical intersections will enrich the understanding of their mechanistic role in molecular photochemistry.

Accidental degeneracy in the spiropyran radical cation: charge transfer between two orthogonal rings inducing ultra-efficient reactivity

We unravel an original photoswitching mechanism in spiropyran radical cation.

Accurate nonadiabatic dynamics

This Perspective addresses the use of coupled diabatic potential energy surfaces (PESs) together with rigorous quantum dynamics in full or reduced dimensional coordinate spaces to obtain accurate solutions to problems in nonadiabatic dynamics.

Assessment of Approximate Coupled-Cluster and Algebraic-Diagrammatic-Construction Methods for Ground- and Excited-State Reaction Paths and the Conical-Intersection Seam of a Retinal-Chromophore Model

As a minimal model of the chromophore of rhodopsin proteins, the penta-2,4-dieniminium cation (PSB3) poses a challenging test system for the assessment of electronic-structure methods for the exploration of ground- and excited-state potential-energy surfaces, the topography of conical intersections, and the dimensionality (topology) of the branching space. Herein, we report on the performance of the approximate linear-response coupled-cluster method of second order (CC2) and the algebraic-diagrammatic-construction scheme of the polarization propagator of second and third orders (ADC(2) and ADC(3)). For the ADC(2) method, we considered both the strict and extended variants (ADC(2)-s and ADC(2)-x). For both CC2 and ADC methods, we also tested the spin-component-scaled (SCS) and spin-opposite-scaled (SOS) variants. We have explored several ground- and excited-state reaction paths, a circular path centered around the S1/S0 surface crossing, and a 2D scan of the potential-energy surfaces along the branching space. We find that the CC2 and ADC methods yield a different dimensionality of the intersection space. While the ADC methods yield a linear intersection topology, we find a conical intersection topology for the CC2 method. We present computational evidence showing that the linear-response CC2 method yields a surface crossing between the reference state and the first response state featuring characteristics that are expected for a true conical intersection. Finally, we test the performance of these methods for the approximate geometry optimization of the S1/S0 minimum-energy conical intersection and compare the geometries with available data from multireference methods. The present study provides new insight into the performance of linear-response CC2 and polarization-propagator ADC methods for molecular electronic spectroscopy and applications in computational photochemistry.

Valence-bond non-equilibrium solvation model for a twisting monomethine cyanine

We propose and analyze a two-state valence-bond model of non-equilibrium solvation effects on the excited-state twisting reaction of monomethine cyanines. Suppression of this reaction is thought responsible for environment-dependent fluorescence yield enhancement in these dyes. Fluorescence is quenched because twisting is accompanied via the formation of dark twisted intramolecular charge-transfer (TICT) states. For monomethine cyanines, where the ground state is a superposition of structures with different bond and charge localizations, there are two possible twisting pathways with different charge localizations in the excited state. For parameters corresponding to symmetric monomethines, the model predicts two low-energy twisting channels on the excited-state surface, which leads to a manifold of TICT states. For typical monomethines, twisting on the excited state surface will occur with a small barrier or no barrier. Changes in the solvation configuration can differentially stabilize TICT states in channels corresponding to different bonds, and that the position of a conical intersection between adiabatic states moves in response to solvation to stabilize either one channel or the other. There is a conical intersection seam that grows along the bottom of the excited-state potential with increasing solvent polarity. For monomethine cyanines with modest-sized terminal groups in moderately polar solution, the bottom of the excited-state potential surface is completely spanned by a conical intersection seam.

Geometric Rotation of the Nuclear Gradient at a Conical Intersection: Extension to Complex Rotation of Diabatic States

Nonadiabatic dynamics in the vicinity of conical intersections is of essential importance in photochemistry. It is well known that if the branching space is represented in polar coordinates, then for a geometry represented by angle θ, the corresponding adiabatic states are obtained from the diabatic states with the mixing angle θ/2. In an equivalent way, one can study the relation between the real rotation of diabatic states and the resulting nuclear gradient. In this work, we extend the concept to allow a complex rotation of diabatic states to form a nonstationary superposition of electronic states. Our main result is that this leads to an elliptical transformation of the effective potential energy surfaces; i.e., the magnitude of the initial nuclear gradient changes as well as its direction. We fully explore gradient changes that result from varying both θ and ϕ (the complex rotation angle) as a way of electronically controlling nuclear motion, through Ehrenfest dynamics simulations for benzene cation.

Ultrafast spectroscopy and structural characterization of a photochromic isomerizing ruthenium bis-sulfoxide complex

Irradiation of [Ru(bpy)2(bpSOp)](PF6)2 (where bpy is 2,2'-bipyridine and bpSOp is 1,3-bis(phenylsulfinyl)propane) results in the formation of two new isomers, namely the S,O- and O,O-bonded species. The crystal structure of the bis-thioether and bis-sulfoxide complexes are reported. NMR spectroscopy of the bis-thioether complex in solution is consistent with the molecular structure determined by diffraction methods. Further, NMR spectroscopy of the bis-sulfoxide complex reveals two conformers in solution, one that is consistent with the solid state structure and a second conformer showing distortion in the aliphatic portion of the chelate ring. Time-resolved visible absorption spectroscopy reveals isomerization time constants of 91 ps in dichloroethane (DCE) and 229 ps in propylene carbonate (PC). Aggregate isomerization quantum yields of 0.57 and 0.42 have been determined in DCE and in PC, respectively. The kinetics of the thermal reversion from the O,O- to S,O-bonded isomer are strongly solvent dependent, occurring with rates of 2.41 × 10(-3) and 4.39 × 10(-5) s(-1) in DCE, and 4.68 × 10(-4) and 9.79 × 10(-6) s(-1) in PC. The two kinetic components are assigned to the two isomers identified in solution.

Semiclassical quantization of nonadiabatic systems with hopping periodic orbits

We present a semiclassical quantization condition, i.e., quantum-classical correspondence, for steady states of nonadiabatic systems consisting of fast and slow degrees of freedom (DOFs) by extending Gutzwiller's trace formula to a nonadiabatic form. The quantum-classical correspondence indicates that a set of primitive hopping periodic orbits, which are invariant under time evolution in the phase space of the slow DOF, should be quantized. The semiclassical quantization is then applied to a simple nonadiabatic model and accurately reproduces exact quantum energy levels. In addition to the semiclassical quantization condition, we also discuss chaotic dynamics involved in the classical limit of nonadiabatic dynamics.

Direct Signature of Light-Induced Conical Intersections in Diatomics

Nonadiabatic effects are ubiquitous in physics, chemistry, and biology. They are strongly amplified by conical intersections (CIs), which are degeneracies between electronic states of triatomic or larger molecules. A few years ago it was revealed that CIs in molecular systems can be formed by laser light, even in diatomics. Because of the prevailing strong nonadiabatic couplings, the existence of such laser-induced conical intersections (LICIs) may considerably change the dynamical behavior of molecular systems. By analyzing the photodissociation process of the D2+ molecule carefully, we found a robust effect in the angular distribution of the photofragments that serves as a direct signature of the LICI, providing undoubted evidence of its existence.

On the formation of anions: frequency-, angle-, and time-resolved photoelectron imaging of the menadione radical anion

Frequency-, angle-, and time-resolved photoelectron imaging of gas-phase menadione (vitamin K3) radical anions was used to show that quasi-bound resonances of the anion can act as efficient doorway states to produce metastable ground electronic state anions on a sub-picosecond timescale. Several anion resonances have been experimentally observed and identified with the assistance of ab initio calculations, and ground state anion recovery was observed across the first 3 eV above threshold. Time-resolved measurements revealed the mechanism of electronic ground state anion formation, which first involves a cascade of very fast internal conversion processes to a bound electronic state that, in turn, decays by slower internal conversion to the ground state. Autodetachment processes from populated resonances are inefficient compared with electronic relaxation through internal conversion. The mechanistic understanding gained provides insight into the formation of radical anions in biological and astrochemical systems.

A theoretical exploration of the nonradiative deactivation of hydrogen-bond complexes: isoindole–pyridine and quinoline–pyrrole

CC2 potential energy profiles of the ground and excited states of the isoindole–pyridine complex along the proton transfer reaction coordinate are studied.

Non-adiabatic dynamics close to conical intersections and the surface hopping perspective

Conical intersections play a major role in the current understanding of electronic de-excitation in polyatomic molecules, and thus in the description of photochemistry and photophysics of molecular systems. This article reviews aspects of the basic theory underlying the description of non-adiabatic transitions at conical intersections, with particular emphasis on the important case when the dynamics of the nuclei are treated classically. Within this classical nuclear motion framework, the main aspects of the surface hopping methodology in the conical intersection context are presented. The emerging picture from this treatment is that of electronic transitions around conical intersections dominated by the interplay of the nuclear velocity and the derivative non-adiabatic coupling vector field.

Photochemistry and photophysics at extended seams of conical intersection

The role of extended seams of conical intersection in excited-state mechanisms is reviewed. Seams are crossings of the potential energy surface in many dimensions where the decay from the excited to the ground state can occur, and the extended seam is composed of different segments lying along a reaction coordinate. Every segment is associated with a different primary photoproduct, which gives rise to competing pathways. This idea is first illustrated for fulvene and ethylene, and then it is used to explain more complex cases such as the dependence of the isomerisation of retinal chromophore isomers on the protein environment, the dependence of the efficiency of the azobenzene photochemical switch on the wavelength of irradiation and the direction of the isomerisation, and the coexistence of different mechanisms in the photo-induced Wolff rearrangement of diazonaphthoquinone. The role of extended seams in the photophysics of the DNA nucleobases and the relationship between two-state seams and three-state crossings is also discussed. As an outlook, the design of optical control strategies based on the passage of the excited molecule through the seam is considered, and it is shown how the excited-state lifetime of fulvene can be modulated by shaping the energy of the seam.

Shape of Multireference, Equation-of-Motion Coupled-Cluster, and Density Functional Theory Potential Energy Surfaces at a Conical Intersection

We report and characterize ground-state and excited-state potential energy profiles using a variety of electronic structure methods along a loop lying on the branching plane associated with a conical intersection (CI) of a reduced retinal model, the penta-2,4-dieniminium cation (PSB3). Whereas the performance of the equation-of-motion coupled-cluster, density functional theory, and multireference methods had been tested along the excited- and ground-state paths of PSB3 in our earlier work, the ability of these methods to correctly describe the potential energy surface shape along a CI branching plane has not yet been investigated. This is the focus of the present contribution. We find, in agreement with earlier studies by others, that standard time-dependent DFT (TDDFT) does not yield the correct two-dimensional (i.e., conical) crossing along the branching plane but rather a one-dimensional (i.e., linear) crossing along the same plane. The same type of behavior is found for SS-CASPT2(IPEA=0), SS-CASPT2(IPEA=0.25), spin-projected SF-TDDFT, EOM-SF-CCSD, and, finally, for the reference MRCISD+Q method. In contrast, we found that MRCISD, CASSCF, MS-CASPT2(IPEA=0), MS-CASPT2(IPEA=0.25), XMCQDPT2, QD-NEVPT2, non-spin-projected SF-TDDFT, and SI-SA-REKS yield the expected conical crossing. To assess the effect of the different crossing topologies (i.e., linear or conical) on the PSB3 photoisomerization efficiency, we discuss the results of 100 semiclassical trajectories computed by CASSCF and SS-CASPT2(IPEA=0.25) for a PSB3 derivative. We show that for the same initial conditions, the two methods yield similar dynamics leading to isomerization quantum yields that differ by only a few percent.

Relative Stability of the La and Lb Excited States in Adenine and Guanine: Direct Evidence from TD-DFT Calculations of MCD Spectra

The relative position of La and Lb ππ* electronic states in purine nucleobases is a much debated topic, since it can strongly affect our understanding of their photoexcited dynamics. To assess this point, we calculated the absorption and magnetic circular dichroism (MCD) spectra of adenine, guanine, and their nucleosides in gas-phase and aqueous solution, exploiting recent developments in MCD computational technology within time-dependent density functional theory. MCD spectroscopy allows us to resolve the intense S0→ La transition from the weak S0→ Lb transition. The spectra obtained in water solution, by using B3LYP and CAM-B3LYP functionals and describing solvent effect by cluster models and by the polarizable continuum model (PCM), are in very good agreement with the experimental counterparts, thus providing direct and unambiguous evidence that the energy ordering predicted by TD-DFT, La < Lb, is the correct one.

Solvent influence on the excited states of the oxo form of barbituric acid and the mechanisms of the out-of-plane non-radiative elongation of the N-H bond: a comparative theoretical and experimental study

The solvent influence on the excited states, emission and absorption energies of the oxo for of barbituric acid was studied with experimental (UV and fluorescence spectra) and theoretical methods. The excited-state reaction paths of the out-pf-plane elongation of the N-H bond of the oxo form of barbituric acid were also investigated (TD DFT level) to the conical intersections mediating internal conversions to the ground state. The (1)nσ* excited state was found to be the driven electronic state. We found that the increase of the polarity of the solvent reduces the (1)nσ* excited state decay rate through a non-planar conical intersection and increases its energy. Thus, solvents with higher polarity disfavor the non-radiative decay through conical intersections.

Dynamical Outcomes of Quenching: Reflections on a Conical Intersection

This review focuses on experimental studies of the dynamical outcomes following collisional quenching of electronically excited OH A2Σ+ radicals by molecular partners. The experimental observables include the branching between reactive and nonreactive decay channels, kinetic energy release, and quantum state distributions of the products. Complementary theoretical investigations reveal regions of strong nonadiabatic coupling, known as conical intersections, which facilitate the quenching process. The dynamical outcomes observed experimentally are connected to the local forces and geometric properties of the nuclei in the conical intersection region. Dynamical calculations for the benchmark OH-H2 system are in good accord with experimental observations, demonstrating that the outcomes reflect the strong coupling in the conical intersection region as the system evolves from the excited electronic state to quenched products.

One photon yields two isomerizations: large atomic displacements during electronic excited-state dynamics in ruthenium sulfoxide complexes

Photochromic compounds efficiently transduce photonic energy to potential energy for excited-state bond-breaking and bond-forming reactions. A critical feature of this reaction is the nature of the electronic excited-state potential energy surface and how this surface facilitates large nuclear displacements and rearrangements. We have prepared two photochromic ruthenium sulfoxide complexes that feature two isomerization reactions following absorption of a single photon. We show by femtosecond transient absorption spectroscopy that this reaction is complete within a few hundred picoseconds and suggest that isomerization occurs along a conical intersection seam formed by the ground-state and excited-state potential energy surfaces.

Low-lying excited-states of 5-benzyluracil

A numerical study is reported concerning the first and second singlet excited-states of 5-benzyluracil using the multireference self-consistent field (state-averaged CASSCF) method. The vertical excitation energies of low-lying excited-states were characterized using the SA-CASSCF method, as well as using higher-level methods, such as CASPT2, MRCI and EOM-CCSD. The local minima and conical intersections found on the potential energy surfaces (PESs) were characterized in terms of molecular geometry and natural population analysis. Different relaxation pathways on the PESs are identified and discussed by comparing with the similar pathways found for the individual monomers of uracil and benzene. The molecule can be thought of as a model system for the study of crosslink reaction between DNA and proteins induced by UV light.

Excited-state tautomerization of gas-phase cytosine

In order to investigate experimentally observed phototautomerization of gas-phase cytosine, several excited-state tautomerization mechanisms were characterized at the EOM-CCSD and TDDFT levels. All pathways that took place exclusively on the S1 surface were found to have significant barriers that were much higher than the barriers involved in radiationless decay of cytosine tautomers through conical intersections back to the ground state; tautomerization in this fashion cannot compete with radiationless relaxation. However, an alternative possibility is that the conical intersections that facilitate radiationless decay could also facilitate tautomerization. Barrierless pathways indicate that it is energetically possible that bifurcation at the conical intersections can lead to a subset of the population reaching different tautomers. This could be an explanation for the observed tautomerization of keto cytosine after exposure to low-energy UV light.

Ultrafast dynamics of aniline following 269-238 nm excitation and the role of the S2(pi3s/pi sigma) state

Femtosecond time-resolved photoelectron imaging is employed to investigate ultrafast electronic relaxation in aniline, a prototypical aromatic amine. The molecule is excited at wavelengths between 269 and 238 nm. We observe that the S2(pi3s/pi sigma*) state is populated directly during the excitation process at all wavelengths and that the population bifurcates to two decay pathways. One of these involves ultrafast relaxation from the Rydberg component of S2(pi3s/pi sigma*) to the S1(Pi Pi)* state, from which it relaxes back to the electronic ground state on a much longer timescale. The other appears to involve motion along the pi sigma* dissociative potential energy surface. At higher excitation energies, the dominant excitation is to the S3(pi pi*) state, which undergoes extremely efficient electronic relaxation back to the ground state. Our study supports some conclusions reached from H-atom photofragment translational spectroscopy measurements and pump-probe photoionization measurements and contradicts some others.

Ultrafast Excited-State Dynamics of ortho-Terphenyl and 1,2-Diphenylcyclohexene: The Role of "Ethylenic Twisting" in the Nonadiabatic Photocyclization of Stilbene Analogs

Nonadiabatic photocyclization is the fundamental step underlying photoswitching and light-assisted bond formation within diarylethylenes, yet the details of the nuclear dynamics leading to cyclization remain unclear. We have examined the ultrafast excited-state dynamics of o-terphenyl (OTP) and 1,2-diphenylcyclohexene (DPCH) in solution to determine how variation in structural constraints impacts the course of nonadiabatic photocyclization specifically in stilbenoids. Measured spectral dynamics reflect cyclization through a S1-to-S0 transition for both systems on picosecond time scales, with excited-state decay appreciably faster for DPCH versus OTP. Supportive ab initio calculations reveal a higher energetic penalty in OTP versus DPCH for reaching the lowest-energy conical intersection from the S1 minimum; this penalty is associated primarily with twisting about the carbon-carbon bond that bridges terminal phenyl groups, a structural change that has a critical role in nonadiabatic cis-trans isomerization of diarylethylenes. Findings provide a new experimental perspective on the elusive nuclear dynamics underlying cis-stilbene photocyclization.

Theoretical methods for ultrafast spectroscopy

Time-resolved spectroscopy in the femtosecond and attosecond time domain is a tool to unravel the dynamics of nuclear and electronic motion in molecular systems. Theoretical insight into the underlying physical processes is ideally gained by solving the time-dependent Schrödinger equation. In this work, methods currently used to solve this equation are reviewed in a compact presentation. These methods involve numerical representations of wavefunctions and operators, the calculation of time evolution operators, the setting up of the Hamiltonian operators and the types of coordinates to be used hereto. The advantages and disadvantages of some methods are discussed.