Handbook of computational chemistry

 From Quantum Theory to Computational Chemistry A Brief Account of Developments Lucjan Piela Department of Chemistry, University of Warsaw, Poland Introduction – Exceptional Status of Chemistry  A Hypothetical Perfect Computer  Does Predicting Mean Understanding?  Orbital Model  Power of Computer Experiments  Conclusions  Acknowledgments  References  J Leszczynski (ed.), Handbook of Computational Chemistry, DOI ./----_, © Springer Science+Business Media B.V    From Quantum Theory to Computational Chemistry A Brief Account of Developments Abstract: Quantum chemical calculations rely on a few fortunate circumstances, like usually small relativistic and negligible electrodynamic (QED) corrections, and large nucleito-electrons mass ratio Unprecedented progress in computer technology has revolutionized quantum chemistry, making it a valuable tool for experimenters It is important for computational chemistry to elaborate methods that look at molecules in a multiscale way, provide its global and synthetic description, and compare this description with those for other molecules Only such a picture can free researchers from seeing molecules as a series of case-by-case studies Chemistry is a science of analogies and similarities, and computational chemistry should provide the tools for seeing this Introduction – Exceptional Status of Chemistry Contemporary science fails to explain the largest-scale phenomena taking place in the universe, such as the speeding up of the galaxies (supposedly due to the undefined “black energy”) and the nature of the lion’s share of the universe’s matter (and also unknown “dark matter”) Quantum chemistry is in a far better position, which may be regarded even as exceptional in the sciences The chemical phenomena are explainable down to individual molecules (which represent the subject of quantum chemistry) by current theories It turned out, by comparing theory and experiment, that the solution to the Schrödinger equation (Schrödinger a, b, c, d) offers in most cases a quantitatively correct picture Only molecules with very heavy atoms, due to the relativistic effects becoming important, need to be treated in a special way based on the Dirac theory (Dirac a, b) This involves an approximate Hamiltonian in the form of the sum of Dirac Hamiltonians for individual electrons, and the electron–electron interactions in the form of the (non-relativistic) Coulomb terms, a common and computationally successful practice ignoring, however, the resulting resonance character of all the eigenvalues (Brown and Ravenhall ; Pestka et al ) When, very rarely, higher accuracy is needed, one may eventually include the quantum electrodynamics (QED) corrections, a procedure currently far from routine application, but still feasible for very small systems (Łach et al ) This success of computational quantum chemistry is based on a few quite fortunate circumstances (for references see, e.g., Piela ): • • • • Atoms and molecules are built of only two kinds of particles: nuclei and electrons Although nuclei have non-zero size (electrons are regarded as point-like particles), the size is so small that its influence is below chemical accuracy (Łach et al ) Therefore, all the constituents of atoms and molecules are treated routinely as point charges The QED corrections are much smaller than energy changes in chemical phenomena (e.g.,  : ) and may be safely neglected in most applications (Łach et al ) The nuclei are thousands times heavier than electrons and therefore, except in some special situations, they move thousands times slower than electrons This makes it possible to solve the Schrödinger equation for electrons, assuming that the nuclei not move, i.e., their positions are fixed in space (“clamped nuclei”) This concept is usually presented within the so called adiabatic approximation In this approximation the motion of the nuclei is considered in the next step, in which the electronic energy (precalculated for any position of the nuclei), together with a usually small diagonal correction for coupling the nuclei-electrons motion, plays the role of the potential energy surface (PES) The total wave From Quantum Theory to Computational Chemistry A Brief Account of Developments  function is assumed to be a product of the electronic wave function and of the function describing the motion of the nuclei The commonly used Born–Oppenheimer (Born and Oppenheimer ) approximation (B-O) is less accurate than the adiabatic one, because it neglects the above-mentioned diagonal correction, making the PES independent of nuclear masses Using the PES concept one may introduce the crucial idea of the spatial structure of a molecule, defined as those positions of the nuclei that assure a minimum of the PES This concept may be traced back to Hund (a, b, c) Moreover, this structure corresponds to a certain ground-state electron density distribution that exhibits atomic cores, atom–atom bonds, and atomic lone pairs It is generally believed that the exact analytical solution to the Schrödinger equation for any atom (except the hydrogen-like atom) or molecule is not possible Instead, some reasonable approximate solutions can be obtained, practically always involving calculation of a large number of molecular integrals, and some algebraic manipulations on matrices built of these integrals The reason for this is efficiency of what is known as algebraic approximation (“algebraization”) of the Schrödinger equation The algebraization is achieved by postulating a certain finite basis set {Φ i }i=M i= and expanding the unknown wave function as a linear combination of the “known” Φ i with unknown expansion coefficients Such an expansion can be encountered in the one-electron case (e.g., linear combination of atomic orbitals introduced by Bloch ), or/and in the many-electron case, e.g., the total wave function expansion in Slater determinants, related to configurations (Slater ), or in the explicitly correlated manyelectron functions (Hylleraas ) It is assumed for good quality calculations (arguments are as a rule of numerical character only) that a finite M chosen is large enough to produce sufficient accuracy, with respect to what would be with M = ∞ (exact solution) The above-mentioned integrals appear because, after the expansion is inserted into the Schrödinger equation, one makes the scalar products (they represent the integrals, which should be easy to calculate) of the expansion with Φ  , Φ  , , Φ M , consecutively In this way the task of finding the wave function by solving the Schrödinger equation is converted into an algebraic problem of finding the expansion coefficients, usually by solving some matrix equation It remains to take care of the choice of the basis set {Φ i } i=M i= The choice represents a technical problem, but unfortunately it contains a lot of arbitrariness and, at the same time, is one of the most important factors influencing cost and quality of the computed solution Application of functions Φ i based on the Gaussian-type one-electron orbitals (GTO) (Boys et al ) provides a low cost/quality ratio and this fact is considered as one of the most important factors that has made computational chemistry so efficient Algebraization involves as a rule a large M and therefore the whole procedure requires fast computing facilities. These facilities changed over time, from very modest manual mechanical calculators at the beginning of the twentieth century to what we consider now as powerful supercomputers Almost immediately after formulation of quantum mechanics in , Douglas Hartree published several papers (Hartree ) presenting his manual calculator-based solutions for atoms of rubidium and chlorine However amazing it looks now, these were self-consistent ab initio computations   Computational chemistry contributed significantly to applied mathematics, because new methods had to be invented in order to treat the algebraic problems of a previously unknown scale (like for M of the order of billions), see, e.g., Roos () That is, derived from the first principles of (non-relativistic) quantum mechanics    From Quantum Theory to Computational Chemistry A Brief Account of Developments In , Walter Heitler and Fritz Wolfgang London clarified the origin of the covalent chemical bond (Heitler and London ), the concept crucial for chemistry In the paper the authors demonstrated, in numerical calculations, that the nature of the covalent chemical bond in H is of quantum character, because the (semiquantitatively) correct description of H emerged only after inclusion the exchange of electrons  and  between the nuclei in the formula a()b() (a, b are the s atomic orbitals centered on nucleus a and nucleus b, respectively) resulting in the wave function a()b()+a()b() Thus, taking into account also the contribution of Hund (a, b, c),  is therefore the year of birth of computational chemistry Perhaps the most outstanding manual calculator calculations were performed in  by Hubert James and Albert Coolidge for the hydrogen molecule (James and Coolidge ) This variational result has been the best one in the literature over a period of  years The s marked the beginning of a new era – the time of programmable computers Apparently, just another tool for number crunching became available In fact, however, the idea of programming made a revolution because it • • • • • • Liberated humans from tedious manual calculations Offered large speed of computation, incomparable to any manual calculator Also, the new data storage tools soon became of massive character Resulted in more and more efficient programs, based on earlier versions (like staying “on the shoulders of the giants”), offering possibilities to calculate dozens of molecular properties Allowed the dispersed, parallel and remote calculations Resulted in the new branch of chemistry: computational chemistry  Allowed performing calculations by anyone, even those not trained in chemistry, quantum chemistry, mathematics, etc The first ab initio Hartree–Fock calculations (based on ideas of Douglas Hartree () and Vladimir Fock (a, b)) on programmable computers for diatomic molecules were performed at the Massachusetts Institute of Technology in , using a basis set of Slater-type orbitals The first calculations with Gaussian-type orbitals were carried out by Boys and coworkers, also in  (Boys et al ) An unprecedented spectroscopic accuracy was obtained for the hydrogen molecule in  by Kołos and Roothaan () In the early s the era of gigantic programs began with the possibility to compute many physical quantities at various levels of approximation We currently live in an era with computational possibilities growing exponentially (the notorious “Moore law” of doubling the computer power every  years). This enormous progress revolutionized our civilization in a very short time The revolution in computational quantum chemistry changed chemistry in general, because computations became feasible for molecules of interest for experimental chemists The progress has been accompanied by achievements in theory, however mainly of the character related to computational needs Today, fairly accurate computations are possible for molecules composed    It is difficult to define what computational chemistry is Obviously, whatever involves calculations in chemistry might be treated as part of it This, however, sounds like a pure banality The same is true with the idea that computational chemistry means chemistry that uses computers It is questionable whether this problem needs any solution at all If yes, the author sticks to the opinion that computational chemistry means quantitative description of chemical phenomena at the molecular level Perhaps the best known is GAUSSIAN, elaborated by a large team headed by John Pople The speed as well as the capacity of computer, memory increased about  billion times over a period of  years This means that what now takes an hour of computations, would require in  about , years of computing From Quantum Theory to Computational Chemistry A Brief Account of Developments  of several hundreds of atoms, spectroscopic accuracy is achievable for molecules with a dozen atoms, while the QED calculations can be performed for the smallest molecules only (few atoms) A Hypothetical Perfect Computer Suppose we have at our disposal a computer that is able to solve the Schrödinger equation exactly for any system and in negligible time. Thus, we have free access to the absolute detailed picture of any molecule This means we may predict with high accuracy and confidence the value of any property of any molecule We might be tempted to say that being able to give such predictions is the ultimate goal of science: “We know everything about our system If you want to know more about the world, take other molecules and just compute, you will know.” Let us consider a system composed of  carbon nuclei,  protons and  electrons Suppose we want to know the geometry of the system for the ground state The computer answers that it does not know what we mean by the term “geometry” We are more precise now and say that we are interested in the carbon–carbon (CC) and carbon–hydrogen (CH) distances The computer answers that it is possible to compute only the mean distances, and provides them together with the proton–proton, carbon–electron, proton–electron and electron–electron distances, because it treats all the particles on an equal footing We look at the CC and CH distances and see that they are much larger than we expected for the CC and CH bonds in benzene The reason is that in our perfect wave function the permutational symmetry is correctly included This means that the average carbon–proton distance takes into account all carbons and all protons The same with other distances To deduce more we may ask for computing other quantities like angles, involving three nuclei Here, too, we will be confronted with numbers including averaging over identical particles These difficulties not necessarily mean that the molecule has no spatial structure at all, although this can also happen The numbers produced would be extremely difficult to translate into a D picture even for quite small molecules, not to mention such a floppy molecule as a protein In many cases we would obtain a D picture we did not expect This is because many molecular structures we are familiar with represent higher-energy metastable electronic states (isomers) This is the case in our example When solving the time-dependent Schrödinger equation, we are confronted with this problem Let us use as a starting wave function the one corresponding to the benzene molecule In time-evolution we will stay probably with a similar geometry for a long time However, there is a chance that after a long period the wave function changes to that corresponding to three interacting acethylene molecules (three times HCCH) The Born–Oppenheimer optimized ground electronic state corresponds to the benzene [−. au in the Hartree–Fock approximation for the --G(d) basis set] The three isolated acethylene molecules (in the same approximation) have the energy −. au, and the molecule (also with the same formula C H ) H C − C ≡ C − C ≡ CH − . au Thus, the benzene molecule seems to be a stable ground-state, while the three acethylenes and the diacethylene are metastable states within the same ground electronic state of the system  In addition, we assume the computer is so clever, that it automatically rejects those solutions, which are not square-integrable or not satisfy the requirements of symmetry for fermions and bosons Thus, all non-physical solutions are rejected    From Quantum Theory to Computational Chemistry A Brief Account of Developments All the three physically observed realizations of the system C + H are separated by barriers; this is the reason why they are observable What is, therefore, the most stable electronic ground state corresponding to the flask of benzene? This is a quite different question, which pertains to systems larger than a single molecule If we multiply the number of atoms in a single molecule of benzene by a natural number N, we are confronted with new possibilities of combining atoms into molecules, not necessarily of the same kind and possibly larger than C H For a large N we are practically unable to find all the possibilities In some cases, when based on chemical intuition and limiting to simple molecules, we may guess particular solutions For example, to lower the energy for the flask of benzene we may allow formation of the methane molecules and the graphite (the most stable form of carbon) Therefore, the flask of benzene represents a metastable state Suppose we wish to know the dipole moment of, say, the HCl molecule, the quantity that tells us important information about the charge distribution We look up the output and we not find anything about dipole moment The reason is that all molecules have the same dipole moment in any of their stationary state Ψ, and this dipole moment equals to zero, see, e.g., Piela () p  Indeed, the dipole moment is calculated as the mean value of the dipole moment operator i.e., μ = ⟨Ψ∣ μˆ Ψ⟩ = ⟨Ψ∣ (∑i q i r i ) Ψ⟩, index i runs over all electrons and nuclei This integral can be calculated very easily: the integrand is antisymmetric with respect to inversion and therefore μ =  Let us stress that our conclusion pertains to the total wave function, which has to reflect the space isotropy leading to the zero dipole moment, because all orientations in space are equally probable If one applied the transformation r → −r only to some particles in the molecule (e.g., electrons), and not to the other ones (e.g., the nuclei), then the wave function will show no parity (it would be neither symmetric nor antisymmetric) We this in the adiabatic or Born–Oppenheimer approximation, where the electronic wave function depends on the electronic coordinates only This explains why the integral μ = ⟨Ψ∣ˆμ Ψ⟩ (the integration is over electronic coordinates only) does not equal zero for some molecules (which we call polar) Thus, to calculate the dipole moment we have to use the adiabatic or the Born–Oppenheimer approximation Now we decide to introduce the Born–Oppenheimer approximation (we resign from the absolute correct picture) and to focus on the most important features of the molecule The first, most natural one, is the molecular geometry, this one that leads to a minimum of the electronic energy The problem is that usually we have many such minima of different energy, each minimum corresponding to its own electronic density distribution Each such distribution corresponds to some particular chemical bonds pattern. In most cases the user of computers does not even think of these minima, because he or she performs the calculations for a predefined configuration of the nuclei and forces the system (usually not being aware of it) to stay in its vicinity This is especially severe for large molecules, such as proteins They have an astronomic number of stable conformations, but often we take one of them and perform the calculations for this one It is difficult to say why we select this one, because we rarely even consider the other conformations In this situation we usually take as the starting point a crystal structure conformation (we believe in its relevance for a free molecule)   Bond patterns are almost the same for different conformers For a dipeptide one has something like ten energy minima, counting only the backbone conformations (and not counting the side chain conformations for simplicity) For a very small protein of, say, a hundred amino acids, the number of conformations is therefore of the order of   , a very large number exceeding the estimated number of atoms in the Universe From Quantum Theory to Computational Chemistry A Brief Account of Developments  Moreover, usually one starts calculations by setting a starting electronic density distribution The choice of this density distribution may influence the final electronic density and the final geometry of the molecule In routine computations one guesses the starting density according to the starting nuclear configuration chosen This may seem to be a reasonable choice, except when small deformation of the nuclear framework leads to large changes in the electronic density In conclusion, in practice the computer gives the solution which is close to what the computing person considers as “reasonable” and sets as the starting point Does Predicting Mean Understanding? The existing commercial programs allow us to make calculations for molecules, treating each molecule as a new task, as if every molecule represented a new world, which has nothing to with other molecules We might not be satisfied with such a picture We might be curious about the following: • • • • • • • • Living in the D space, does the system have a certain shape or not? If yes, why the shape is of this particular kind? Is the shape fairly rigid or rather flexible? Are there some characteristic substructures in the system? How they interact? How they influence the calculated global properties, etc? Are the same substructures present in other molecules? Does the presence of the same substructures determine similar properties? It is of fundamental importance for chemistry that we not study particular cases, case by case, but derive some general rules Strictly speaking these rules are false because, due to approximations made, they are valid to some extent only However, despite this, they enable chemists to operate, to understand, and to be efficient If we relied uniquely on exact solutions of the Schrödinger equation, there would be no chemistry at all; people would lose the power of rationalizing chemistry, in particular to design syntheses of new molecules Chemists rely on molecular spatial structure (nuclear framework), on the concepts of valence electrons, chemical bonds, electronic lone pairs, importance of HOMO and LUMO energies, etc All these notions have no rigorous definition, but they still are of great importance in describing a model of molecule A chemist predicts that two OH bonds have similar properties, wherever they are in molecule Moreover, chemists are able to predict differences in the OH bonds by considering what the neighboring atoms are in each case It is of fundamental importance in chemistry that a group of atoms with a certain bond pattern (functional group) represents an entity that behaves similarly, when present in different molecules We have at our disposal various scales at which we can look at details of the molecule under study In the crudest approach we may treat the molecule as a point mass, which contributes to the gas pressure Next we might become interested in the shape of the molecule, and we may approximate it first as a rigid rotator and get an estimation of rotational levels we can expect Then we may leave the rigid body model and allow the atoms of the molecule to vibrate about their equilibrium positions In such a case we need to know the corresponding force constants This requires either choosing the structural formula (chemical bond pattern) of the molecule    From Quantum Theory to Computational Chemistry A Brief Account of Developments together with taking the corresponding empirical force constants, or applying the normal mode analysis, first solving the Schrödinger equation in the Born–Oppenheimer approximation (we have a wide choice of the methods of solution) In the first case, we obtain an estimation of the vibrational levels, in the second, we get more reliable vibrational analysis, especially for larger atomic orbital expansions If we wish we may consider anharmonicity of vibrations. At the same time we obtain the electronic density distribution from the wave function Ψ for N electrons  ρ(r) = N ∑σ =  ∫ dτ  dτ  dτ N ∣Ψ(r, σ , r , σ , , r N , σ N )∣   According to the Hellmann–Feynman theorem (Feynman ; Hellmann ), ρ is sufficient to compute the forces acting on the nuclei We may compare the resulting ρ calculated at different levels of approximation, and even with the naive structural formula The density distribution ρ can be analyzed in the way known as Bader analysis (Bader ) First, we find all the critical points, in which ∇ρ =  Then, one analyzes the nature of each critical point by diagonalizing the Hessian matrix calculated at the point : • • • • If the three eigenvalues are negative, the critical point corresponds to a maximum of ρ If two are negative and one positive, the critical point corresponds to a covalent bond If one is negative and two positive, the critical point corresponds to a center of an atomic ring If all three are positive, the critical point corresponds to an atomic cavity The chemical bond critical points correspond to some pairs of atoms; there are other pairs of atoms, which not form bonds The Bader analysis enables chemists to see molecules in a synthetic way, nearly independent of the level of theory that has been used to describe it, focusing on the ensemble of critical points We may compare this density with the density of other molecules, similar to ours, to see whether one can note some local similarities We may continue this, getting a more and more detailed picture down to the almost exact solution of the Schrödinger equation It is important in chemistry to follow such a way, because at its beginning as well as at its end we know very little about chemistry We learn chemistry on the way  The low-frequency vibrations may be used as indicators to look at possible instabilities of the molecule, such as dissociation channels, formation of new bonds, etc Moving all atoms, first according to a lowfrequency normal mode vibration and continuing the atomic displacements according to the maximum gradient decrease, we may find the saddle point, and then, sliding down, detect the products of a reaction channel  The integration of ∣Ψ∣ is over the coordinates (space and spin ones) of all the electrons except one (in our case the electron  with the coordinates r, σ  ) and in addition the summation over its spin coordinate (σ  ) As a result one obtains a function of the position of the electron  in space: ρ(r) The wave function Ψ is antisymmetric with respect to exchange of the coordinates of any two electrons, and, therefore, ∣Ψ∣ is symmetric with respect to such an exchange Hence, the definition of ρ is independent of the label of the electron we not integrate over According to this definition, ρ represents nothing else but the density of the electron cloud carrying N electrons, and is proportional to the probability density of finding an electron at position r  Strictly speaking the nuclear attractors not represent critical points, because of the cusp condition (Kato )  We may also analyze ρ using a “magnifying glass” represented by −Δρ From Quantum Theory to Computational Chemistry A Brief Account of Developments  Orbital Model The wave function for identical fermions has to be antisymmetric with respect to exchange of coordinates (space and spin ones) of any two of them This means that two electrons having the same spin coordinate cannot occupy the same position in space Since wave functions are continuous this means that electrons of the same spin coordinate avoid each other (“Fermi hole” or “exchange hole” about each of them) This Pauli exclusion principle does not pertain to two electrons of opposite spin However, electrons repel one another (Coulombic force) at any finite distance, i.e., have to avoid one another because of their charge (“Coulomb hole” or “correlation hole” around each of them) It turned out, references in Piela () p , that the Fermi hole is by far more important than the Coulomb hole A high-quality wave function has to reflect the Fermi and the Coulomb holes The corresponding mathematical expression should have the antisymmetrization operator in front, this will take care of the Pauli principle (and introduce a Fermi hole) Besides this, it should have some parameters or mathematical structure controlling somehow the distance between any pair of electrons (this will introduce the Coulomb repulsion) Since the Fermi hole is much more important, it is reasonable to consider first a wave function that takes care of the Fermi hole only The simplest way to take the Fermi hole into account is the orbital model (approximation) Within the orbital model the most advanced is the Hartree–Fock method In this method the Fermi hole is taken into account by construction (antisymmetrizer) The Coulomb hole is not present, because the Coulomb interaction is calculated through averaging the positions of the electrons The orbital model is wrong, because it neglects the Coulomb hole Being wrong, it has however, enormous scientific power, because: • • • • • • • It allows one to see the electronic structure as contributions of individual electrons, with their own “wave functions” i.e., orbitals with a definite mathematical form, symmetry, energy (“orbital levels”), etc We take the Pauli exclusion principle into account by not allowing occupations of an orbital by more than two electrons (if two, then of the opposite spin coordinates) The occupation of all orbital levels is known as orbital diagram The orbital energy may be interpreted as the energy needed to remove an electron from the orbital (assuming that all the orbitals not change during the removing, Koopmans’ theorem, Koopmaans ) Molecular electron excitations may often be identified with changing the electron occupancy in the orbital diagram We may even consider electron correlation (Coulomb hole), either by allowing different orbitals for electrons of different spin, or considering a wave function expansion composed of electron diagrams with various occupations One may trace the molecular perturbations to changes in the orbital diagram One may describe chemical reactions as a continuous change from a starting to a final molecular diagram Theory and computational experience bring some rules, like that only those orbitals of the molecular constituents mix, which have similar orbital energies and have the same symmetry This leads to important symmetry selection rules for chemical reactions (Fukui and Fujimoto ; Woodward and Hoffmann ) and for optical excitations (Cotton )    Ab Initio Investigation of Photochemical Reaction Mechanisms reduction of the full opsin model to a model with only the  amino acids of Palczewski’s cavity (Palczewski et al ), those that surround the retinal It was found that the value predicted for the S → S excitation energy is substantially the same indicating that for Rh the charges of distant residues are of minor importance for the optical absorption features of PSB Of course, we not expect this to be a general result Highly truncated protein models should always be applied carefully until a systematic analysis is carried out to determine their limitations and reliability In > Fig - top left, we report the S branch of the photochemical reaction path of Rh computed in terms of a relaxed scan driven by the torsional deformation of the reacting bond Using this data we have located and assigned the S structure that corresponds to transient fluorescent state I We then located the structure of the lowest lying S /S CI (CIRh) that also corresponds to the local S minimum This structure displays a ∼○ twisted C–C bond Starting from the CI-Rh geometry, using standard optimization, the first stable ground state intermediate bathorhodopsin (batho-Rh) has been located It exhibits absorption at  nm and features an all-trans-like chromophore structure (∼○ dihedral angle around the C–C double bond) The computed structure of batho-Rh can be compared with the structure that was experimentally derived by Mathies and coworkers (Kukura et al ) by femtosecond resolved resonance Raman spectroscopy (see > Fig - top right) The comparison indicates that it is possible to predict the structure of an intermediate Concerning the photon energy of ∼ kcal ⋅ mol− that is efficiently stored in batho-Rh, the results of our CASPT//CASSCF/AMBER energy profile show that we reproduce this quantity with an error of  kcal ⋅ mol− A mechanistic picture of the space-saving isomerization mechanism is derived from the reaction path of the Rh QM/MM model and the associated structural deformation The predominant change in the retinal geometry that occurs immediately after excitation leading to the fluorescent state I is a bond length alteration (BLA) in the –C = C–C = C–C = C– moiety A complete inversion between single and double bonds is found at the geometry characterizing the state I The CI-Rh displays a highly helical structure compared with Rh and FS-Rh and is mainly characterized by a large structural change in the –C = C–C = C–C = C– moiety Thus, the motion driving the S → S decay is mainly torsional with a rotation of ∼○ (○ → ○ ) around the C = C bond, and ○ and ○ twisting around the C = C and C = C bonds, respectively Therefore, from a general point of view, during photoisomerization the structural changes not occur exclusively at the central double bond but also involve the other two adjacent double bonds, which lead to a global change in the helicity of the chromophore A mechanism can be derived considering the largest changes These involve the torsion about the reactive bond C = C and about the adjacent C = C bond As highlighted in > Fig - (bottom) these twisting deformations occur in opposite directions and results in the rotation of the –CH–CH– moiety with respect to the remaining framework The mechanism confirms that the space-saving motion imposed by the tight Rh cavity is of the crankshaft type However, since the progression about the C = C bond is more limited, we can talk of an asynchronous crankshaft mechanism (Frutos et al ) Furthermore, a comparison of the CI-Rh structure with the batho-Rh structure establishes (consistent with the experimental data) that the less twisted C = C reverts to its original stereochemistry after ground state relaxation, so we can talk of an aborted asynchronous crankshaft mechanism Another widely studied retinal protein is bacteriorhodopsin (bR), a light-driven ion pump discovered in Halobacterium salinarum in  (Oesterhelt and Stoeckenius ) It is a membrane protein, which upon illumination generates and maintains a proton gradient across the Ab Initio Investigation of Photochemical Reaction Mechanisms  cell membrane (Oesterhelt and Stoeckenius ) The gradient can be used as a source of energy, for example, by adenosine triphosphate synthase bR shares some similarities with the visual rhodopsin: seven transmembrane α-helices and a protonated Schiff base with lysinebound retinal as the chromophore The X-ray crystal structure was first obtained at . Å resolution in  (Grigorieff et al ) By , the resolution had improved to . Å (Luecke et al ) The bR photocycle consists of several intermediates labeled K, L, M, N, and O, which were characterized by spectroscopy and trapped at low temperature in order to be studied by crystallography (Balashov and Ebrey ; Lanyi ) Light absorption triggers the Z/E isomerization of the C = C double bond of the PSBAT chromophore The bond isomerization is completed at the K intermediate, yielding a -cis chromophore (PSB); the following thermal relaxation leads to Schiff base proton transfer to the Asp counter-ion, in the L and M steps The proton flux is completed during the remaining part of the photocycle, along with other accompanying proton transfers and protein structural changes The first ab initio QM/MM calculation on bR was reported by Hayashi et al using multiconfigurational methods to evaluate the excited state properties (Hayashi and Ohmine ) The protein model was based on the crystal structure determined by Luecke et al () where the missing residues were taken from the PDB structures BRD (Grigorieff et al ), CW, (Luecke et al ) and QHJ (Belrhali et al ) The geometry was optimized at the HF level, using DZV and -G basis sets with polarization functions on oxygen atoms CASSCF(,) was employed to calculate retinal excitation energies Five different models were built, each with different atoms included in the QM region The results demonstrated that the protein environment affects the absorption properties of the chromophore The side chains and the water molecules in the binding pocket affect the geometry of the retinal, forcing a twisting of the C = C double bond The protein environment is anionic near the Schiff base, which stabilizes the ground state with respect to the excited state, causing a blue shift of the S → S excitation energy Later the Schulten group (Hayashi et al ) performed a further analysis to elucidate the physical mechanisms of the observed spectroscopic tuning in the rhodopsin family A comparison between bR and the sensory rhodopsin II (sRII) was done to understand the origin of the a change in λ max from  nm in bR to  nm in SRII This difference was observed experimentally, in spite of the same chromophore in both proteins and the similar protein environments As in the previous work, a QM/MM methodology was employed, with AMBER as the MM force field The equilibrium geometries of the QM regions were computed at the HF/DZV level The excitation energies were evaluated with a CASSCF(,) wave function The QM region included the retinal, the retinal-bound lysine (starting from Cγ), and parts of the binding pocket: two aspartic acid residues (starting from C β ) and three water molecules The comparison between bR and SRII was performed by calculating the contributions to the S –S excitation energy: the total energy was rewritten as a sum of the energy of the isolated chromophore, the electronic reorganization energy arising from the modification of the wave function in the presence of the protein, and the electrostatic interaction energy between protein and chromophore The results showed that the main contribution is electronic reorganization in the retinal and a closer inspection of the structures showed a reduced counter-ion–Schiff base distance in sRII compared with bR This is the reason for the blue shift, consistent with the picture given in a previous paper (Hayashi et al ) Another spectroscopic feature of the bR photocycle was addressed by the same group in  (Hayashi et al ) The early K and L intermediate models were built using MD and refined with QM/MM optimization, following the same method as described above The    Ab Initio Investigation of Photochemical Reaction Mechanisms geometry optimization showed an increased chromophore distortion on passing from the ground state to the K intermediate Moreover, the local hydrogen bond network is perturbed: a water molecule is displaced and the hydrogen bond between Thr and Asp is broken because the latter group undergoes a conformational change Another result of that change is the reduction of the distance between the Asp carboxyl group and the Schiff base, which is thought to prompt the proton transfer in the following L-to-M process Regarding the optical properties, K and L intermediates are red-shifted with respect to the initial absorption of bR The excitation energies were evaluated at the CASSCF(,)//HF level, with a three state averaged calculation followed by state specific computation for S and S The theoretical results are in good agreement with the experimental red shift, even though the absolute values are quite different and the spectroscopic shifts are overestimated The reason for the shortcomings of the computational theory in this case is the failure of the CASSCF method to account for dynamic correlation An analysis of the individual contributions to the total excitation energy allows one to pinpoint the main factors affecting the spectroscopic shift Chromophore distortion around the C = N and C = C double bonds was found to be the main factor It destabilizes S more than S , therefore reducing the energy gap Another factor is the reduced electrostatic interaction between protein and chromophore, which increases the S energy without affecting S , where the charge is more delocalized towards the β-ionone ring Nevertheless the geometry is optimized at the HF level, which is known to overestimate bonding character, compared with post-SCF methods, which leads to a likely underestimation of bond torsion in the chromophore Deactivation Mechanism in Cytosine-Guanine DNA Base Pair Deoxyribonucleic acid (DNA) is a common biomolecule in all living organisms that contains the genetic information used for their development and functioning It is composed of nucleotides, with backbones made of sugars and phosphate groups joined by ester bonds, forming long polymer chains Attached to each sugar is one of only four bases: adenine (A), cytosine (C), guanine (G), and thymine (T) These bases belong to two types of organic molecules – adenine and guanine are purines that are fused five- and six-membered heterocyclic compounds, while cytosine and thymine are pyrimidines, which are six-membered rings A pair of strands is usually arranged in the shape of a double helix by strong hydrogen bonds between the bases that stack upon each other Due to the aromatic character of the nucleotide bases DNA absorbs in the harmful ultraviolet (UV) region of the spectrum Hence, UV irradiation is one sort of mutagen that can damage DNA and subsequently lead to cancer It is not surprising that DNA and its building blocks have been extensively investigated by computational quantum chemistry The full coverage of the applications goes beyond the scope of this book chapter Instead we are reporting an investigation using multiconfigurational ab initio calculations combined with AMBER force field in a QM/MM approach to study ultrafast radiationless deactivation mechanism of cytosine-guanine inter-strand base pair (Groenhof et al ) Groenhof et al () used a multiconfigurational approach as part of the QM/MM setup to study the inter-strand excited state proton transfer One of the fastest deactivation processes in DNA was related to the cytosine-guanine base pair occurring on sub-picosecond time-scale Therefore excited state molecular dynamics were employed to track this reaction CASSCF with a reduced active space of eight electrons in eight orbitals and a -G basis set was employed to model the molecular dynamics of the photoactivated C–G base pair A surface hopping algorithm was used to detect a transition from the excited to the ground state Ab Initio Investigation of Photochemical Reaction Mechanisms  The crystal structure of the human DNA/topoisomerase I complex provided the initial coordinates for the QM/MM simulations In total,  base pairs of B-DNA molecule were used The partitioning of the system was done in a way that the cytosine-guanine base pair in the center of the molecule was described at the QM level The remainder of the system was modeled with AMBER force field (Case et al ) The chemical bonds between the bases and the deoxyribose sugar rings connecting the QM and the MM subsystems were replaced by constraints and the QM part was capped with two hydrogen link atoms To equilibrate the DNA and the solvent prior to the QM/MM simulations, the system was equilibrated classically for , ps The initial conditions for the excited-state simulations were obtained by taking  frames at equal time intervals from an additional  ps ground state trajectory at the CASSCF(,) level In the excited state MD simulations, a time step of . fs was used In consequence, the ultrafast photodeactivation mechanism of the cytosine-guanine base pair was uncovered by the molecular dynamics simulations (see > Fig -) The excitation to the charge transfer state induces a transfer of a proton from guanine to cytosine within a few femtoseconds After the proton transfer, the system approaches the conical intersection seam and returns to the ground state However, within a few femtoseconds the seam is closed again, and a second hop takes the system back to S where it stays until another hop occurs Most trajectories showed several hopping events between the S and S surfaces, with an average excited state lifetime of  fs for the entire process This can be rationalized by the topology of the S –S intersection space (see > Fig -) The proton transfer coordinate in the vicinity of the S minimum is parallel to the extended hyperline and allows multiple transitions between the crossing states S1 t = 31 fs t = 32 fs t = 44 fs proton transfer S1 / S0 conical intersection S0 cytosine t = 200 fs guanine t = 82 fs t = 54 fs double proton transfer ⊡ Fig - Sequence of events after the excitation A proton is abstracted from guanine and accepted by cytosine After the decay to the ground state via a conical intersection the proton returns to the guanine base The excess thermal energy that is released upon returning to S is responsible for spontaneous double proton transfer (at  fs) which leads to the formation of a different tautomeric state of the base pair ( fs) (Redrawn with permission from ref Groenhof et al )   Ab Initio Investigation of Photochemical Reaction Mechanisms S1 S1/ S0 seam S0 FC region S0 pro ton tra nsf er de ske fo let rm al at ion S1 hν energy  ⊡ Fig - Excited and ground state potential energy surface crossing of the cytosine-guanine base pair The two reaction coordinates are proton transfer and skeletal deformation of the bonds The dashed yellow and green lines schematically represent a path sampled in a typical trajectory on the S and the S potential, respectively Along the proton-transfer coordinate the system moves out of Franck–Condon region to the minimum of the excited state Due to the oscillations of the second reaction coordinate the trajectory hits the seam more than once (Redrawn with permission from ref Groenhof et al ) Absorption Spectra of a Coumarin in Solution The calculation of absorption spectra is of great interest to chemists, since these spectra provide a wealth of information about the molecule, its environment and its properties As a result of solvent effects and solute-solvent interactions the spectra can be quiet complex In addition, some of the information can be convoluted under the broad bands Nowadays, computational photochemistry serves as a standard tool for spectroscopists, used to assign the experimental spectra Until recently the calculations yield simple stick absorption spectra, which show the excitation energies and their relative intensities, without taking environmental effects into account Recent advances in quantum chemistry allow researchers to treat solvents and to calculate vibrational contributions to the spectra Here we present a case study of coumarin C (> Scheme -) computed by TD-DFT including solvent effects considered by polarized continuum model (PCM) Improta, Barone, and Santoro have studied the spectra in two different solvents: cyclohexane and dimethylsulfoxide (DMSO) (Improta et al ) Coumarin C was selected to evaluate this novel methodology because it exhibits significantly different polarizabilities on its ground and electronically excited states Therefore solvents with different polarities lead to large solvatochromic shifts that ought to be accounted for by the simulations The calculated stick absorption spectrum of the anti isomer of coumarin at  K in cyclohexane is shown in > Fig - left The authors employ an algorithm for automatic selection of transitions between vibrational states since the full set is too computationally demanding After harmonic analysis the algorithm was used to select ⋅ transitions out of a total of  Ab Initio Investigation of Photochemical Reaction Mechanisms  ⊡ Scheme - Cyclohexane DMSO V70 V18 V86 V55 V88 24000 26000 28000 30000 Frequency / cm–1 32000 0-0 C1 Absorption Absorption Absorption 0-0 C2 C3 24000 26000 C4 28000 Frequency / cm–1 30000 32000 20000 22000 24000 26000 28000 30000 Frequency / cm–1 ⊡ Fig - Calculated spectra of C in cyclohexane Left:  K stick spectrum and its assignment including transitions with one quantum in a single oscillator Middle: A decomposition of the spectrum (thick line) in its components, where Cn dominates collection of transitions to vibrational states with the same number n of simultaneously excited oscillators Right: Calculated spectra in cyclohexane (blue) and DMSO (red) compared to the experimental spectra (black) (Redrawn with permission from ref Improta et al ) states For the convolution of the spectrum the full width at half maximum of a Gaussian was chosen to match the main experimental bands The spectrum was found to be unaltered when calculated for  K and also to be practically indistinguishable for the anti and syn isomers The line shapes of the calculated spectrum resemble its experimental counterpart as shown for anti coumarin in cyclohexane (> Fig - middle) The two-peak structure with a spacing of , cm− is qualitatively reproduced The first band is mainly due to the – transition and a contribution from transitions to the single vibrationally excited state (C) However, the computed spectrum is found to be blue-shifted by  cm− The second band is composed of contributions from transitions to the first (C), second (C), and third (C) vibrational states The only deviation from the experimental spectrum is manifested in the relative height of these two bands In DMSO the spectrum consisting of an asymmetric broad band, extended towards the blue wing of the maximum, is correctly reproduced by the simulation (> Fig - right) This proves the reliability of the calculated vibrational progression hidden within the band However, similar to the case of cyclohexane, the spectrum is found to be red-shifted by cm− The authors ascribe the latter to the effect of dynamic solvent fluctuations which are much more pronounced in polar solvents Finally, the DMSO→cyclohexane solvatochromic shift is estimated to be only  cm− smaller than its experimental counterpart    Ab Initio Investigation of Photochemical Reaction Mechanisms Using this novel protocol Improta, Barone, and Santoro were able to simulate the spectrum of coumarin C in different solution environments with a high degree of accuracy Complex band structures were reproduced for solvents of varying polarities with a computational procedure feasible for large molecular systems Excited State Molecular Dynamics Tracking the Photoisomerization of Retinal in Different Environments The first attempts to study the photoisomerization of the retinal chromophore by using ab initio excited state molecular dynamics were reported by Robb and coworkers (Vreven et al ; Weingart et al ) focusing on a model chromophore in the gas phase at the CASSCF level of theory A more recent work by Weingart, again for an isolated retinal model, has shown that the quantum yield of the photoisomerization process depends critically on the initial retinal configuration (Weingart ) In the resting state of bovine rhodopsin, the chromophore deviates from planarity (Okada et al ; Palczewski et al ) By taking this retinal structure as a starting point for a series of excited state CASSCF semi-classical trajectories the author found that % of the trajectories ended at the all-trans photoproduct This result suggests that selectivity and quantum efficiency might originate from strain imposed by steric interactions between the chromophore and the protein before photon absorption Recently we have reported the first S trajectory computation for Rh carried out with a scaled CASSCF force field that reproduces the static and transient spectroscopic parameters (Frutos et al ) Because of the immense cost of the CASPT gradients, their application in trajectory computation is impossible for the retinal chromophore even if such a calculation is limited to a few hundred femtoseconds However, CASSCF energies not only fail to reproduce the observed spectroscopic properties of Rh but also yield an energy profile that is too steep In order to overcome this problem and evaluate a realistic trajectory we noticed that scaling the CASSCF isomerization energy profile along the S branch of the reaction path of > Fig - top left, yields a curve overlapping with the corresponding CASPT curve (Frutos et al ) As displayed in > Fig - bottom left, the results show that this potential drives the chromophore to the CI on a  fs time scale This time scale is in line with the observed excited state Rh lifetimes, which confirms the suitability of the scaled force field The assertion that a scaled CASSCF potential can approximate CASPT accuracy has been assessed by single point CASPT//CASSCF/AMBER calculations along the trajectory In > Fig - bottom-left we report the S potential energy along the computed trajectory Seven single point CASPT/-G∗ /AMBER computations have been performed to validate the scaled energy profile and to compute the correct S –S energy gap The scaled CASSCF energies remain close to the CASPT energies all along the trajectory supporting the accuracy of our procedure Within  fs the S system undergoes a ∼ kcal ⋅ mol− energy decrease Following this event the potential energy decreases slowly and monotonically until, after  fs, a region of degeneracy located ∼ kcal ⋅ mol− below the Franck–Condon point is reached Since the initial motion is dominated by simultaneous double bond expansion and single bond contraction,  kcal ⋅ mol− of vibrational energy must be initially located along this mode, which leads Ab Initio Investigation of Photochemical Reaction Mechanisms  to complete inversion of the double bond/single bond character centered in the –C = C– C = C–C = C– moiety The analysis of the entire  fs S trajectory together with the resulting backbone deformation depicted in > Fig -, points to a space saving isomerization mechanism that includes the previously proposed bicycle pedal or crankshaft coordinate The – C = C–C = C–C = segment of PSB twists with respect to the two remaining fragments, namely the –NH = C–C = C–C = and the β-ionone ring, during the first – fs Such a motion is mainly characterized by a negative twist of the reactive –C = C– bond and a positive twist of the –C–C– bond At the critical  fs mark the nature of the motion changes The –C–C– twist stops and the –C = C bond adjacent to the reactive –C = C– bond starts to twist in the positive direction In other words, the = C–C = fragment rotates with respect to the backbone, leading to an ○ twisted –C = C– bond and to a moderately twisted (∼○ ) –C = C– bond after  fs The nature of this motion is confirmed by plotting the linear momentum vectors in the – fs region Further studies have revealed that this motion may be an intrinsic property of the retinal chromophore (Schapiro et al ) A more recent CASSCF/AMBER trajectory study of the photoisomerization of Rh was reported by Hayashi et al () The protein model was constructed based on the U crystal structure It was equilibrated using classical MD and provided, after a  ns equilibration, a sample of  different starting points for the trajectory calculations The polyene chain of the retinal, including the double bond of the β-ionone ring, was treated at the CASSCF level using a DZV basis set The active space was composed of  π-electrons and  π-type orbitals The first two roots were averaged in the CASSCF wave function In order to describe the interaction between the QM part and the MM part, mechanical embedding was employed This is an important difference from the excited state trajectory presented above (Frutos et al ) that uses electrostatic embedding instead In mechanical embedding the interaction between the QM and the MM region is treated at the MM level, which is less accurate However, the transition from the excited to the ground state was initiated using the energy difference as criterion, namely when it was less than . kcal ⋅ mol− In addition, a trajectory of isolated retinal was calculated using a structure taken from the equilibrated rhodopsin It was found that all  CASSCF/AMBER trajectories decayed to the ground state within  fs (the shorter time could be attributed to a lack of scaling and the unrealistically steep CASSCF energy surface), which is of the same order of magnitude as the trajectory by Frutos et al () Batho-Rh is formed in  cases and in one case a -cis isomer (isorhodopsin) is produced by a bicycle pedal isomerization of C–C and C–C bond However, an accompanying rotation of ○ , on average, is found around C–C in the  trajectories showing a crankshaft or aborted bicycle pedal-type mechanism, which is consistent with the previous findings The first QM/MM trajectory computation describing the photoisomerization of bR was reported by Hayashi et al () However, the selected QM part corresponded to a highly reduced PSBAT chromophore featuring only three double bonds This was described by a three state averaged CASSCF wave function with  π-electrons in a  π-type orbitals active space with a DZV basis set The MM part was described using AMBER for the protein and the TIPP model for the water molecules An ensemble of starting conditions was generated by a classical MD simulation of the resting state of bR at constant temperature ( K) To compare the effect of the protein on the photoisomerization, trajectories of the same retinal model were calculated under vacuum In total,  trajectories were performed to see if the high selectivity of the isomerization could be reproduced in the absence of the protein environment It was found that the initial stretching relaxation is followed by a selective isomerization exclusively around    Ab Initio Investigation of Photochemical Reaction Mechanisms the C–C bond It should be mentioned that this double bond is the central one in the investigated model Hence, its preference for rotation compared with the terminal bonds is expected The analysis of the isomerization dynamics of the six isolated chromophore trajectories shows that four trajectories isomerize around C–C and two around the terminal C–N bond The authors concluded that the high selectivity in bR is a result of the protein environment Conclusion In this chapter we have presented a few selected computational photochemistry case studies of model systems with increasing complexity, starting from isolated molecules that served as models in early works The photochemical reaction paths of these systems were examined in mechanistic investigations, resulting in a general understanding of light-induced processes Towards the end of the chapter we have given more recent examples of steady state spectra, minimum energy path calculations, and molecular dynamics simulations in different environments More realistic models and rigorous simulation not only provided an explanation for the photochemical events, but also allowed a comparison with experimental data: the shape of spectra in solution can be convoluted, ab initio dynamics can predict excited state lifetimes and quantum yields, and conical intersections of chromophores in proteins can be located All this reveals that the current stage of the tools of computational photochemistry are already providing results with accuracy comparable to experimental observables Nevertheless, we have also shown that some questions remain open, needing more accurate treatments which are still not feasible to date Therefore further breakthroughs in computational photochemistry are anticipated On one hand the development of quantum chemical methods and algorithms for effective computations is expected in the near future, whereas an increase of computational resources and computing power on the other hand ought to lead to another bloom in this research field References Adams, T R., Adamson, R D., Austin, B., Baker, J., Bell, A T., Beran, G J O., Baer, R., Besley, N A., Brooks, B., Brown, S T., Byrd, E C., Casanova, D., Challacombe, M., Chai, J., Chakraborty, A K., Cheng, C., Chien, S., Chipman, D M., Cramer, C., Crittenden, D., Dachsel, H., Jr., R A D., Doerksen, R J., Dombroski, J P., Dreuw, A., Dunietz, B D., Dutoi, A D., Florian, J., Furlani, T R., Epifanovsky, E., Gan, Z., Ghysels, A., Gilbert, A T B., Gill, P M W., Head-Gordon, M., Gwaltney, S R., Hawkins, G., Hehre, W J., Herbert, J M., Heyden, A., Hohenstein, E G., Hirata, S., Hsu, C., Ishikawa, N., Johnson, B G., Jung, Y., Kedziora, G., Keil, F J., Kelly, C., Khaliullin, R Z., Klunzinger, P., Korambath, P P., Krylov, A I., Kurlancheek, W., Kong, J., Kussmann, J., Lee, A M., Lee, M., Levchenko, S V., Liang, W., Lin, C Y., Liotard, D., Livshits, E., Lochan, R C., Lotan, I., Manohar, P., Marenich, A., Maslen, P E., Maurice, D R., Fusti-Molnar, L., Nair, N., Oana, M., O’Neill, D P., Neuhauser, D., Ochsenfeld, C., Olson, R., Oumi, M., Peters, B., Pieniazek, P A., Proynov, E I., Rassolov, V A., Rhee, Y., Ritchie, J P., Rohrdanz, M A., Rosta, E., Schaefer, H F., III , Shao, Y., Sherrill, C D., Schultz, N E., Simmonett, A C., Slipchenko, L V., Sodt, A., Steele, R P., Subotnik, J E., Thanthiriwatte, K., Thom, A J W., Truhlar, D G., Vanovschi, V., Voorhis, T V., Wang, T., Warshel, A., White, C., Williams, C F., Lee Woodcock, H., Wu, Q., You, Z., Zhang, W., & Zhao, Y () Q-Chem (. ed.) Pittsburgh: Q-Chem, Inc Altoe, P., Stenta, M., & Garavelli, M () Rhodopsin and GFP chromophores: QM/MM absorption spectra in solvent and protein AIP Conference Proceedings, (), – Ab Initio Investigation of Photochemical Reaction Mechanisms Andersson, K., Aquilante, F., Barysz, M., Bednarz, E., Bernhardsson, A., Blomberg, M R A., Carissan, Y., Cooper, D L., Cossi, M., Devarajan, A., Vico, L D., Ferré, N., Fülscher, M P., Gaenko, A., Gagliardi, L., Ghigo, G., de Graaf, C., Hess, B A., Hagberg, D., Holt, A., Karlström, G., Krogh, J W., Lindh, R., Malmqvist, P-Å., Nakajima, T., Neogrády, P., Olsen, J., Pedersen, T B., Raab, J., Reiher, M., Roos, B O., Ryde, U., Schimmelpfennig, B., Schütz, M., Seijo, L., Serrano-Andrés, L., Siegbahn, P E M., Stålring, J., Thorsteinsson, T., Veryazov, V., Widmark, P.-O., & Wolf, A () MOLCAS (Version . ed.) Lund: University of Lund Andersson, K., Malmqvist, P A., Roos, B O., Sadlej, A J., & Wolinski, K () nd Order perturbation-theory with a CASSCF reference function Journal of Physical Chemistry, (), – Andruniow, T., & Olivucci, M PSBAT absorption spectra in vacuo [Unpublished results] Andruniow, T., & Pawlikowski, M () On the Franck-Condon effects in the absorption spectrum of CH anion The analysis bsed on the ab initio MCSCF method Acta Physica Polonica A, , – Anglada, J M., & Bofill, J M () A reducedrestricted-quasi-Newton-Raphson method for locating and optimizing energy crossing points between two potential energy surfaces Journal of Computational Chemistry, (), – Aquilante, F., De Vico, L., Ferré, N., Ghigo, G., Malmqvist, P.-Å., Neogrády, P., Pedersen, T B., Pitoák, M., Reiher, M., Roos, B O., SerranoAndrés, L., Urban, M., Veryazov, V., & Lindh, R () MOLCAS : The next generation Journal of Computational Chemistry, (), – doi:./jcc. Balashov, S P., & Ebrey, T G () Trapping and spectroscopic identification of the photointermediates of bacteriorhodopsin at low temperatures Photochem Photobiol, (), – Barbatti, M., & Lischka, H () Nonadiabatic deactivation of H-adenine: A comprehensive picture based on mixed quantumclassical dynamics Journal of the American Chemical Society, (), – doi: /jap Barbatti, M., Ruckenbauer, M., Szymczak, J J., Aquino, A J A., & Lischka, H () Nonadiabatic excited-state dynamics of polar pi-systems and related model compounds of biological relevance Physical Chemistry Chemical Physics, (), – doi:./bm  Bearpark, M J., Robb, M A., & Schlegel, H B () A direct method for the location of the lowest energy point on a potential surface crossing Chemical Physics Letters , (), – Belrhali, H., Nollert, P., Royant, A., Menzel, C., Rosenbusch, J P., Landau, E M., & PebayPeyroula, E () Protein, lipid and water organization in bacteriorhodopsin crystals: A molecular view of the purple membrana at . Angstrom resolution Structure (London, England : ), (), – Ben-Nun, M., & Martinez, T J () Nonadiabatic molecular dynamics: Validation of the multiple spawning method for a multidimensional problem Journal of Chemical Physics, (), – Berman, H M., Westbrook, J., Feng, Z., Gilliland, G., Bhat, T N., Weissig, H., Shindyalov, I N., & Bourne, P E () The protein data bank Nucleic Acids Research, (), – Bernardi, F., De, S., Olivucci, M., & Robb, M A () The mechanism of ground-stateforbidden photochemical pericyclic reactions: Evidence for real conical intersections Journal of the American Chemical Society, (), – Bernardi, F., Olivucci, M., Michl, J., & Robb, M A (a) Conical intersections in the theory of organic singlet photochemistry The Spectrum, , – Bernardi, F., Olivucci, M., & Robb, M A (b) Potential energy surface crossings in organic photochemistry Chemical Society Reviews, (), – Bernardi, F., Olivucci, M., & Schegel, H B () Michael Alfred Robb: A short biography Molecular Physics, (), – doi:./  Blomgren, F., & Larsson, S () Exploring the potential energy surface of retinal, a comparison of the performance of different methods Journal of Computational Chemistry, (), – Braslavsky, S E () Glossary of terms used in photochemistry, rd edition Pure and Applied Chemistry, , – Bravaya, K., Bochenkova, A., Granovsky, A., & Nemukhin, A () An opsin shift in rhodopsin: Retinal S-S excitation in protein, in solution, and in the gas phase Journal of the American Chemical Society, (), – doi:./ja Brejc, K., Sixma, T K., Kitts, P A., Kain, S R., Tsien, R Y., Ormö, M., & Remington, S J () Structural basis for dual excitation and photoisomerization of the Aequorea victoria green fluorescent protein Proceedings of the National Academy of    Ab Initio Investigation of Photochemical Reaction Mechanisms Sciences of the United States of America, (), – Brooks, R., Bruccoleri, R E., Olafson, B D., States, D J., Swaminathan, S., & Karplus, M () CHARMM: A program for macromolecular energy, minimization, and dynamics calculations Journal of Computational Chemistry, , – Case, D A., Pearlman, D A., Caldwell, J W., Cheatham, T E., III, Wang, J., Ross, W S., Simmerling, C L., Darden, T A., Merz, K M., Stanton, R V., Cheng, A L., Vincent, J J., Crowley, M., Tsui, V., Gohlke, H., Radmer, R J., Duan, Y., Pitera, J., Massova, I., Seibel, G L., Singh, U C., Weiner, P K., & Kollman, P A () AMBER (th ed.) San Francisco: University of California Cembran, A., Bernardi, F., Olivucci, M., & Garavelli, M () The retinal chromophore/chloride ion pair: Structure of the photoisomerization path and interplay of charge transfer and covalent states Proceedings of the National Academy of Sciences of the United States of America, (), – doi:./pnas  Chattoraj, M., King, B A., Bublitz, G U., & Boxer, S G () Ultra-fast excited state dynamics in green fluorescent protein: Multiple states and proton transfer Proceedings of the National Academy of Sciences of the United States of America, (), – De Vico, L., Olivucci, M., & Lindh, R () New general tools for constrained geometry optimizations Journal of Chemical Theory and Computation, (), – doi:./ct Dreuw, A., & Head-Gordon, M () Singlereference ab initio methods for the calculation of excited states of large molecules Chemical Reviews, (), – Fantacci, S., Migani, A., & Olivucci, M () CASPT//CASSCF and TDDFT//CASSCF mapping of the excited state isomerization path of a minimal model of the retinal chromophore The Journal of Physical Chemistry B, (), –  doi:./jp Farazdel, A., & Dupuis, M () On the determination of the minimum on the crossing seam of two potential energy surfaces Journal of Computational Chemistry, (), – Ferré, N., & Olivucci, M (a) The amide bond: Pitfalls and drawbacks of the link atom scheme Theochem-Journal of Molecular Structure, , – doi:./s-()-  Ferré, N., & Olivucci, M (b) Probing the rhodopsin cavity with reduced retinal models at the CASPT//CASSCF/AMBER level of theory Journal of the American Chemical Society, (), – doi:./jad Ferré, N., Cembran, A., Garavelli, M., & Olivucci, M () Complete-active-space selfconsistent-field/Amber parameterization of the Lys-retinal-Glu rhodopsin chromophore-counterion system Theoretical Chemistry Accounts, (), – doi: /s---  Frisch, M J., Trucks, G W., Schlegel, H B., Scuseria, G E., Robb, M A., Cheeseman, J R., Scalmani, G., Barone, V., Mennucci, B., Petersson, G A., Nakatsuji, H., Caricato, M., Li, X., Hratchian, H P., Izmaylov, A F., Bloino, J., Zheng, G., Sonnenberg, J L., Hada, M., Ehara, M., Toyota, K., Fukuda, R., Hasegawa, J., Ishida, M., Nakajima, T., Honda, Y., Kitao, O., Nakai, H., Vreven, T., Montgomery, J., Jr., Peralta, J E., Ogliaro, F., Bearpark, M., Heyd, J J., Brothers, E., Kudin, K N., Staroverov, V N., Kobayashi, R., Normand, J., Raghavachari, K., Rendell, A., Burant, J C., Iyengar, S S., Tomasi, J., Cossi, M., Rega, N., Millam, J M., Klene, M., Knox, J E., Cross, J B., Bakken, V., Adamo, C., Jaramillo, J., Gomperts, R., Stratmann, R E., Yazyev, O., Austin, A J., Cammi, R., Pomelli, C., Ochterski, J W., Martin, R L., Morokuma, K., Zakrzewski, V.G., Voth, G A., Salvador, P., Dannenberg, J J., Dapprich, S., Daniels, A D., Farkas, O., Foresman, J B., Ortiz, J V., Cioslowski, J., & Fox, D J () GAUSSIAN  (Vol Revision A.) Wallingford CT: Gaussian Inc Frutos, L M., Andruniów, T., Santoro, F., Ferré, N., & Olivucci, M () Tracking the excited-state time evolution of the visual pigment with multiconfigurational quantum chemistry Proceedings of the National Academy of Sciences of the United States of America, (), – doi: /pnas. Gonzalez-Luque, R., Garavelli, M., Bernardi, F., Merchan, M., Robb, M A., & Olivucci, M () Computational evidence in favor of a two-state, two-mode model of the retinal chromophore photoisomerization Proceedings of the National Academy of Sciences of the United States of America, (), – Gordon, M S., Schmidt, M W., Frenking, G., Kim, K S., & Scuseria, G E () Advances in electronic structure theory: GAMESS a decade later In C Dykstra, G Frenking, K S Kim, & G E Scuseria (Eds.), Theory and applications of computational chemistry: The first forty years (pp –) Amsterdam: Elsevier Grabowski, Z R., Rotkiewicz, K., & Rettig, W () Structural changes accompanying Ab Initio Investigation of Photochemical Reaction Mechanisms intramolecular electron transfer: Focus on twisted intramolecular charge-transfer states and structures Chemical Reviews, (), – doi:./crl Grigorenko, B., Savitsky, A., Topol, I., Burt, S., & Nemukhin, A () Ground-state structures and vertical excitations for the kindling fluorescent protein asFP The Journal of Physical Chemistry B, (), – Grigorieff, N., Ceska, T A., Downing, K H., Baldwin, J M., & Henderson, R () Electroncrystallographic refinement of the structure of bacteriorhodopsin Journal of Molecular Biology, , – Groenhof, G., Bouxin-Cademartory, M., Hess, B., deVisser, S P., Berendsen, H J C., Olivucci, M., Mark, A E., & Robb, M A () Photoactivation of the photoactive yellow protein: Why photon absorption triggers a trans-to-cis isomerization of the chromophore in the protein Journal of the American Chemical Society, (), – Groenhof, G., Schäfer, L V., Boggio-Pasqua, M., Goette, M., Grubmuller, H., & Robb, M A () Ultrafast deactivation of an excited cytosine-guanine base pair in DNA Journal of the American Chemical Society, (), –  doi:./jac Groenhof, G., Schäfer, L V., Boggio-Pasqua, M., Grubmueller, H., & Robb, M A () Arginine controls the photoisomerization process in photoactive yellow protein Journal of the American Chemical Society, , – Hayashi, S., & Ohmine, I () Proton transfer in bacteriorhodopsin: Structure, excitation, IR spectra, and potential energy surface analyses by an ab initio QM/MM method The Journal of Physical Chemistry B, (), – Hayashi, S., Tajkhorshid, E., Pebay-Peyroula, E., Royant, A., Landau, E M., Navarro, J., & Schulten, K () Structural determinants of spectral tuning in retinal proteinsbacteriorhodopsin vs sensory rhodopsin II The Journal of Physical Chemistry B, (), – Hayashi, S., Tajkhorshid, E., & Schulten, K () Structural changes during the formation of early intermediates in the bacteriorhodopsin photocycle Biophysical Journal, (), – Hayashi, S., Tajkhorshid, E., & Schulten, K () Molecular dynamics simulation of bacteriorhodopsin’s photoisomerization using ab initio forces for the excited chromophore Biophysical Journal, (), – Hayashi, S., Taikhorshid, E., & Schulten, K () Photochemical reaction dynamics of the primary  event of vision studied by means of a hybrid molecular simulation Biophysical Journal, (), – Hoffmann, M., Wanko, M., Strodel, P., Konig, P H., Frauenheim, T., Schulten, K., Thiel, W., Tajkhorshid, E., & Elstner, M () Color tuning in rhodopsins: The mechanism for the spectral shift between bacteriorhodopsin and sensory rhodopsin II Journal of the American Chemical Society, (), – Improta, R., Barone, V., & Santoro, F () Ab initio calculations of absorption spectra of large molecules in solution: Coumarin C Angewandte Chemie International Edition, (), – Jorgensen, W L., & Tirado-Rives, J () The OPLS [optimized potentials for liquid simulations] potential functions for proteins, energy minimizations for crystals of cyclic peptides and crambin Journal of the American Chemical Society, (),– doi:./jaa Kandori, H., Schichida, Y., & Yoshisawa, T () Photoisomerization in rhodopsin BiochemistryMoscow, , – Koga, N., & Morokuma, K () Determination of the lowest energy point on the crossing seam between two potential surfaces using the energy gradient Chemical Physics Letters, (), – Kukura, P., McCamant, D W., Yoon, S., Wandschneider, D B., & Mathies, R A () Structural observation of the primary isomerization in vision with femtosecond-stimulated Raman Science, (), – doi:./science  Kutateladze, A G () Computational methods in photochemistry Boca Raton: Taylor & Francis Laino, T., & Passerone, D () Pseudo-dynamics and band optimizations: Shedding light into conical intersection seams Chemical Physics Letters, (–), – doi:./j.cplett... Lanyi, J K () Molecular mechanism of ion transport in bacteriorhodopsin: Insights from crystallographic, spectroscopic, kinetic, and mutational studies The Journal of Physical Chemistry B, , – Levine, B G., Ko, C., Quenneville, J., & Martinez, T J () Conical intersections and double excitations in time-dependent density functional theory Molecular Physics, (–), – doi:./ Lischka, H., Shepard, R., Shavitt, I., Pitzer, R M., Dallos, M., Müller, T., Szalay, P G., Brown, F B., Ahlrichs, R., Böhm, H J., Chang, A., Comeau, D C., Gdanitz, R., Dachsel, H., Ehrhardt, C.,    Ab Initio Investigation of Photochemical Reaction Mechanisms Ernzerhof, M., Höchtl, P., Irle, S., Kedziora, G., Kovar, T., Parasuk, V., Pepper, M J M., Scharf, P., Schiffer, H., Schindler, M., Schüler, M., Seth, M., Stahlberg, E A., Zhao, J.-G., Yabushita, S., Zhang, Z., Barbatti, M., Matsika, S., Schuurmann, M., Yarkony, D R., Brozell, S R., Beck, E V., & Blaudeau, J.-P () COLUMBUS, an ab initio electronic structure program (release .. ed.) Vienna: University of Vienna Luecke, H., Schobert, B., Richter, H T., Cartailler, J P., & Lanyi, J K () Structure of bacteriorhodopsin at . Angstrom resolution Journal of Molecular Biology, , – Martin, M E., Negri, F., & Olivucci, M () Origin, nature, and fate of the fluorescent state of the green fluorescent protein chromophore at the CASPT//CASSCF resolution Journal of the American Chemical Society, (), – Mathies, R A., & Lugtenburg, J () The primary photoreaction of rhodopsin In D G de Stavenga, W J & Grip, E N Pugh (Eds.), Handbook of biological physics (Vol , pp –) Amsterdam: Elsevier Science Press Migani, A., & Olivucci, M () A computational strategy for organic photochemistry In W Domcke, D R Yarkony, H Köppel (Eds.), Conical intersections: Electronic structure, dynamics and spectroscopy (pp –) Singapore: World Scientific Migani, A., Robb, M A., & Olivucci, M () Relationship between photoisomerization path and intersection space in a retinal chromophore model Journal of the American Chemical Society, , – Migani, A., Sinicropi, A., Ferré, N., Cembran, A., Garavelli, M., & Olivucci, M () Structure of the intersection space associated with Z/E photoisomerization of retinal in rhodopsin proteins Faraday Discussion, , – Nielsen, I B., Lammich, L., & Andersen, L H () S- and S- excited states of gas-phase schiff-base retinal chromophores Physical Review Letters, (), – Oesterhelt, D., & Stoeckenius, W () Rhodopsinlike protein from the purple membrane of halobacterium halobium Nature New Biology, , – Oesterhelt, D., & Stoeckenius, W () Functions of a new photoreceptor membrane Proceedings of the National Academy of Sciences of the United States of America, , – Okada, T., Sugihara, M., Bondar, A.-N., Elstner, M., Entel, P., & Buss, V () The retinal conformation and its environment in rhodopsin in light of a new . Å crystal structure Journal of Molecular Biology, , – Olivucci, M., & Sinicropi, A () Computational photochemistry In M Olivucci (Ed.), Computational photochemistry Theoretical and computational chemistry (Vol , pp –) Amsterdam: Elsevier Page, C S., & Olivucci, M () Ground and excited state CASPT geometry optimizations of small organic molecules Journal of Computational Chemistry, (), – doi:./jcc  Palczewski, K., Kumasaka, T., Hori, T., Behnke, C A., Motoshima, H., Fox, B A., Trong, I L., Teller, D C., Okada, T., Stenkamp, R E., Yamamoto, M., & Miyano, M () Crystal structure of rhodopsin: A G protein-coupled receptor Science, ,  Pistolesi, S., Sinicropi, A., Pogni, R., Basosi, R., Ferré, N., & Olivucci, M () Modeling the fluorescence of protein-embedded tryptophans with ab initio multiconfigurational quantum chemistry: The limiting cases of parvalbumin and monellin The Journal of Physical Chemistry B, (), – Ponder, J W., & Case, D A () Force fields for protein simulations Advances in Protein Chemistry, , – Quillin, M L., Anstrom, D A., Shu, X K., O’Leary, S., Kallio, K., Chudakov, D A., & Remington, S J () Kindling fluorescent protein from Anemonia sulcata: Dark-state structure at . angstrom resolution Biochemistry-US, (), – Ragazos, I N., Robb, M A., Bernardi, F., & Olivucci, M () Optimization and characterization of the lowest energy point on a conical intersection using an MC-SCF Lagrangian Chemical Physics Letters, (), – Rajput, J., Rahbek, D B., Andersen, L H., Hirshfeld, A., Sheves, M., Altoe, P., Orlandi, G., & Garavelli, M () Probing and modeling the absorption of retinal protein chromophores in vacuo Angewandte Chemie International Edition, (), – doi:./anie. Robb, M A., Bernardi, F., & Olivucci, M () Conical intersection as a mechanistic feature of organic-photochemistry Pure and Applied Chemistry, (), – Roos, B O () The complete active space selfconsistent field method and its applications in electronic structure calculations In K P Lawley (Ed.), Advances in chemical physics; Ab initio methods in quantum chemistry - II (pp –) Chincester: Wiley Ab Initio Investigation of Photochemical Reaction Mechanisms Roos, B O., & Taylor, P R () A complete active space SCF (CASSCF) method using a densitymatrix formulated super-CI approach Chemical Physics, (), – Schäfer, L V., Groenhof, G., Klingen, A R., Ullmann, G M., Boggio-Pasqua, M., Robb, M A., & Grubmueller, H () Photoswitching of the fluorescent protein asFP: Mechanism, proton pathways, and absorption spectra Angewandte Chemie International Edition, (), – Schäfer, L V., Groenhof, G., Boggio-Pasqua, M., Robb, M A., & Grubmuller, H () Chromophore protonation state controls photoswitching of the fluoroprotein asFP PLoS Computational Biology, (), e Schapiro, I., Weingart, O., & Buss, V () Bicycle-pedal isomerization in a rhodopsin chromophore model Journal of the American Chemical Society, (), – doi:./ jaz Schlegel, H B., & Robb, M A () MC SCF gradient optimization of the H  CO→ H  + CO transition structure Chemical Physics Letters, , – Schmidt, M W., & Gordon, M S () The construction and interpretation of MCSCF wavefunctions Annual Review of Physical Chemistry, , – Schmidt, M W., Baldridge, K K., Boatz, J A., Elbert, S T., Gordon, M S., Jensen, J H., Koseki, S., Matsunaga, N., Nguyen, K A., Su, S J., Windus, T L., Dupuis, M., & Montgomery, J A () General atomic and molecular electronic-structure system Journal of Computational Chemistry, (), – Sekharan, S., Weingart, O., & Buss, V () Ground and excited states of retinal Schiff base chromophores by multiconfigurational perturbation theory Biophysical Journal, (), L–L Senn, H M., & Thiel, W () QM/MM methods for biomolecular systems Angewandte Chemie International Edition, (), – Shao, Y., Molnar, L F., Jung, Y., Kussmann, J., Ochsenfeld, C., Brown, S T., Gilbert, A T B., Slipchenko, L V., Levchenko, S V., O’Neill, D P., DiStasio, R A., Lochan, R C., Wang, T., Beran, G J O., Besley, N A., Herbert, J M., Lin, C Y., Van Voorhis, T., Chien, S H., Sodt, A., Steele, R P., Rassolov, V A., Maslen, P E., Korambath, P P., Adamson, R D., Austin, B., Baker, J., Byrd, E F C., Dachsel, H., Doerksen, R J., Dreuw, A., Dunietz, B D., Dutoi, A D., Furlani, T R., Gwaltney, S R., Heyden, A., Hirata, S., Hsu, C P., Kedziora, G., Khalliulin, R Z., Klunzinger, P., Lee, A M., Lee, M S., Liang, W., Lotan, I., Nair,  N., Peters, B., Proynov, E I., Pieniazek, P A., Rhee, Y M., Ritchie, J., Rosta, E., Sherrill, C D., Simmonett, A C., Subotnik, J E., Woodcock, H L., Zhang, W., Bell, A T., Chakraborty, A K., Chipman, D M., Keil, F J., Warshel, A., Hehre, W J., Schaefer, H F., Kong, J., Krylov, A I., Gill, P M W., & Head-Gordon, M () Advances in methods and algorithms in a modern quantum chemistry program package Physical Chemistry Chemical Physics, (), – Shimomura, O () Discovery of green fluorescent protein (GFP) (nobel lecture) Angewandte Chemie International Edition, (), – doi:./anie. Sicilia, F., Blancafort, L., Bearpark, M J., & Robb, M A () Quadratic description of conical intersections: Characterization of critical points on the extended seam The Journal of Physical Chemistry A, (), – doi:./ jpw Sicilia, F., Blancafort, L., Bearpark, M J., & Robb, M A () New algorithms for optimizing and linking conical intersection points Journal of Chemical Theory and Computation, (), – doi:./ct Siegbahn, P E M., Almlof, J., Heiberg, A., & Roos, B O () The complete active space SCF (CASSCF) method in a Newton-Raphson formulation with application to the HNO molecule Journal of Chemical Physics, (), – Simons, J () Energetic principles of chemical reactions Boston: Jones and Bartlett Sinicropi, A () The mechanism of light energy wastage and exploitation in n,p∗ chromophores Ph.D., University of Siena, Siena Sinicropi, A., Pischel, U., Basosi, R., Nau, W M., & Olivucci, M () Conical intersections in charge-transfer induced quenching Angewandte Chemie International Edition, (), – Sinicropi, A., Pogni, R., Basosi, R., Robb, M A., Gramlich, G., Nau, W M., & Olivucci, M () Fluorescence quenching via sequential hydrogen, electron, and proton transfer In the proximity of a conical intersection Angewandte Chemie International Edition, (), – Sinicropi, A., Nau, W M., & Olivucci, M () Excited state quenching via “unsuccessful” chemical reactions Photochemical and Photobiological Sciences, (), – Sinicropi, A., Page, C S., Adam, W., & Olivucci, M () Computational study on the origin of the stereoselectivity for the photochemical denitrogenation of diazabicycloheptene Journal of the American Chemical Society, (), –  doi:./ja    Ab Initio Investigation of Photochemical Reaction Mechanisms Sinicropi, A., Andruniow, T., Ferre, N., Basosi, R., & Olivucci, M () Properties of the emitting state of the green fluorescent protein resolved at the CASPT//CASSCF/CHARMM level Journal of the American Chemical Society, (), – Solomon, B S., Thomas, T F., & Steel, C () Primary processes in the photochemistry of byciclic azo compounds Journal of the American Chemical Society, , – Strambi, A., Coto, P B., Ferré, N., & Olivucci, M () Effects of water re-location and cavity trimming on the CASPT//CASSCF/AMBER excitation energy of rhodopsin Theoretical Chemistry Accounts, (), – doi:./ s--- y Teller, D C., Okada, T., Behnke, C A., Palczewski, K., & Stenkamp, R E () Advances in determination of a high-resolution three-dimensional structure of rhodopsin, a model of G-proteincoupled receptors (GPCRs) Biochemistry-US, , – Turro, N J., Ramamurthy, V., & Scaiano, J C () Principles of molecular photochemistry: An introduction Sausalito: University Science Books Valsson, O., & Filippi, C () Photoisomerization of model retinal chromophores: Insight from quantum monte carlo and multiconfigurational perturbation theory Journal of Chemical Theory and Computation, (), – doi:./ cty Virshup, A M., Punwong, C., Pogorelov, T V., Lindquist, B A., Ko, C., & Martinez, T J () Photodynamics in complex environments: Ab initio multiple spawning quantum mechanical/molecular mechanical dynamics The Journal of Physical Chemistry B, (), – doi:./jp Vreven, T., Bernardi, F., Garavelli, M., Olivucci, M., Robb, M A., & Schlegel, H B () Ab initio photoisomerization dynamics of a simple retinal chromophore model Journal of the American Chemical Society, (), – Wanko, M., Garavelli, M., Bernardi, F., Niehaus, T A., Frauenheim, T., & Elstner, M () A global investigation of excited state surfaces within time-dependent density-functional response theory Journal of Chemical Physics, (), – Wanko, M., Hoffmann, M., Strodel, P., Koslowski, A., Thiel, W., Neese, F., Frauenheim, T., Elstner, M () Calculating absorption shifts for retinal proteins: Computational challenges The Journal of Physical Chemistry B, (), – doi:./jp Warshel, A () Bicycle-pedal model for the first step in the vision process Nature, (), – Warshel, A., & Barboy, N () Energy storage and reaction pathways in the first step of the vision process Journal of the American Chemical Society, (), – Warshel, A., & Chu, Z T () Nature of the Surface Crossing Process in bacteriorhodopsin: Computer simulations of the quantum dynamics of the primary photochemical event The Journal of Physical Chemistry B, , – Warshel, A., & Levitt, M () Theoretical studies of enzymic reactions: Dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme Journal of Molecular Biology, (), – Warshel, A., Chu, Z T., & Hwang, J.-K () The dynamics of the primary event in rhodopsins revisited Chemical Physics,  – Weingart, O () The twisted C = C bond of the rhodopsin chromophore–a photochemical hot spot Journal of the American Chemical Society, (), – doi:./jat Weingart, O., Migani, A., Olivucci, M., Robb, M A., Buss, V., & Hunt, P () Probing the photochemical funnel of a retinal chromophore model via zero-point energy sampling semiclassical dynamics The Journal of Physical Chemistry A, (), – doi:./jpb Werner, H.-J., Knowles, P J., Lindh, R., Manby, F R., Schuetz, M., Celani, P., Korona, T., Mitrushenkov, A., Rauhut, G., Adler, T B., Amos, R D., Bernhardsson, A., Berning, A., Cooper, D L., Deegan, M J O., Dobbyn, A J., Eckert, F., Goll, E., Hampel, C., Hetzer, G., Hrenar, T., Knizia, G., Koeppl, C., Liu, Y., Lloyd, A W., Mata, R A., May, A J., McNicholas, S J., Meyer, W., Mura, M E., Nicklass, A., Palmieri, P., Pflueger, K., Pitzer, R., Reiher, M., Schumann, U., Stoll, H., Stone, A J., Tarroni, R., Thorsteinsson, T., Wang, M., & Wolf, A () MOLPRO, a package of ab initio programs (Version . ed.) Cardiff: Unversity of Cardiff Yamamoto, N., Olivucci, M., Celani, P., Bernardi, F., & Robb, M A () An MC-SCF/MP study of the photochemistry of ,-diazabicyclo [..] hept--ene: Production and fate of diazenyl and hydrazonyl biradicals Journal of the American Chemical Society, (), – Yarkony, D R () Systematic determination of intersections of potential energy surfaces using a Lagrange multiplier constrained procedure Journal of Physical Chemistry, (), – ... Quantum Theory to Computational Chemistry A Brief Account of Developments Lucjan Piela Department of Chemistry, University of Warsaw, Poland Introduction – Exceptional Status of Chemistry ... (ed.), Handbook of Computational Chemistry, DOI ./----_, © Springer Science+Business Media B.V    From Quantum Theory to Computational Chemistry A Brief Account of Developments... series of case-by-case studies Chemistry is a science of analogies and similarities, and computational chemistry should provide the tools for seeing this Introduction – Exceptional Status of Chemistry