摘要:
Paper Calculation of anharmonic vibrational spectroscopy of small biological molecules R. B. Gerber,*a,b B. Brauer,a S. K. Gregurickc and G. M. Chaband aDepartment of Physical Chemistry and Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel bDepartment of Chemistry, University of California, Irvine, CA 92697, USA cDepartment of Chemistry and Biochemistry, University of Maryland, Baltimore County, Baltimore, MD 20215, USA dNASA Ames Research Center, Moffet Field, CA 94035, USA Received 15th August 2002, Accepted 23rd September 2002 First published as an Advance Article on the web 15th October 2002 The role of anharmonic effects in the vibrational spectroscopy of small biological molecules and their 1 : 1 complexes with water is discussed. The strengths and limitations of the vibrational self-consistent field (VSCF) method and its extensions as a computational tool for this purpose are examined.Anharmonic coupling between different vibrational modes is found to be very important for these systems, even for fundamental transitions, and incorporation of these effects seems essential for quantitative interpretation of experimental data. Both analytical, empirical force fields, and potential surfaces computed from electronic structure methods are used in VSCF calculations of several benchmark systems and compared with experimental spectroscopic data. Glycine in several conformers, the glycine–water complex, and N-methylacetamide are among the systems discussed. The main conclusions are: (1) Electronic structure methods such as MP2/DZP and density functional B97, give very good agreement with experimental data.Thus, MP2 and B97 clearly provide an accurate description of the anharmonic interactions. VSCF calculations, including all modes, with MP2, B97 and other successful methods are presently feasible for molecules with up to 15-20 atoms. (2) The electronic structure methods seem to give spectroscopic predictions in much better accord with experiment than standard empirical force fields such as AMBER or OPLS. The anharmonic couplings provided by these methods differ greatly, in the cases tested to date, from the ab initio ones. The implications of these results for future modeling of small biomolecules are discussed. Comments are provided on future directions in this subject, including extensions to large biomolecules.I. Introduction Vibrational spectroscopy is among the foremost experimental tools in the exploration of molecular potential energy surfaces. The application of this method to biological molecules has been limited so far, both by experimental difficulties and by unavailability of adequate computational tools of quantitative interpretation. Recently, however, there has been major progress on both the experimental and theoretical fronts of this problem. An important, long-standing technique for obtaining spectroscopic information on small biological molecules has been matrix spectroscopy.1 Matrix perturbation effects upon the molecular spectroscopy are often moderate, though for many biological molecules the spectra are not sufficiently wellresolved.An important recent advance has been the spectroscopic study of biological molecules and their complexes by jet-expansion techniques.2,3 The merit of these methods is their true isolation of the species of interest in the gas phase. Furthermore, these techniques provide effective temperatures of only a few degrees Kelvin, which is essential for high resolution. Vibrational spectroscopic data for both ground and excited electronic states are obtained by these novel approaches. Another new, emerging technique measures the spectroscopy of biological molecules in superfluid helium droplets.4 This provides an ultracold environment (the temperature is around 0.4 K). Also, due to the low polarizability of helium, the vibrational spectra of the immersed species are expected to be only very slightly perturbed.5 With these novel experimental techniques, availability of high-resolution spectroscopic data is expected to increase dramatically in the coming years.Theoretically, until very recently, calculations of vibrational spectroscopy of biological molecules were restricted to the harmonic approximation.6–8 Such studies have been very useful, but they are also of limited significance since many biological molecules are very ‘‘floppy’’ and subject to strong anharmonic effects. The anharmonic effects are even much larger in weakly-bound complexes of biological molecules, e.g. their hydrogen-bonded complexes with water, which are of great interest in the field. Also, much of the interest in the potential energy surfaces is away from the equilibrium con- figuration, where the harmonic approximation is less applicable.The challenge of anharmonic spectroscopy calculations is substantial, since beyond the harmonic approximation different vibrational modes are not mutually separable. One is faced with the task of calculating wavefunctions and energy levels for systems of many coupled degrees of freedom. The present paper overviews progress in this area. The main method discussed will be the vibrational self-consistent field (VSCF) approach and its extensions,9,10 which so far, appear to be the main tool of anharmonic spectroscopy calculations for biological molecules. The focus will be, however, on the applications and on the early conclusions that seem to emerge as to properties of the potential surfaces of small biological molecules, e.g.amino acids, amino acid/water complexes and peptides. The importance of anharmonic effects, and the adequacy of empirical and ab initio potentials in describing such effects will be central to the discussion. The structure of this review is as follows. Section II briefly describes the methodology. Section III describes the theoretical basis of the CC-VSCF method. Direct calculation Perspective 142 PhysChemComm, 2002, 5(21), 142–150 DOI: 10.1039/b208000a This journal is # The Royal Society of Chemistry 2002of anharmonic vibrational spectroscopy from ab initio potentials is described in Section IV. Section V considers ‘‘benchmark’’ applications and examines the implications for the potential surfaces.Section VI comments on likely future directions. II. Methods The harmonic approximation is often useful for rough estimation of the fundamental transitions in many small biological molecules, but is inadequate for more quantitative purposes. For example, anharmonic calculations that will be discussed in more detail later, show that deviations of 200–300 cm21 of the harmonic results from the experimental ones for the fundamental transitions of the OH and NH stretching vibrations are common.11 For low frequency transitions the absolute errors are, of course, much smaller but the percentage errors in the case of very floppy modes can be extremely large.11 (a) Scaling procedures To bring computed frequencies into closer agreement with experiment, scaling procedures are sometimes applied.12 These use essentially empirical prescriptions to predict anharmonic frequencies from computed harmonic values.12 Useful as such scaling approaches may be in the analysis of experimental data, they do not address the challenge of computing the observable from an anharmonic potential energy surface.In fact, a proper calculation of the latter type should make it possible to learn about the anharmonic part of the potential by attempting to reproduce experimental data. (b) Quantum perturbation theory A relatively simple approach to the calculation of anharmonic spectra is by applying quantum perturbation theory, as done by Bludsky et al.13 in the case of glycine. In this approach, the harmonic approximation is used for the zeroth-order Hamiltonian, and the anharmonic terms of the potential function are treated as the perturbation.The order of perturbation theory used in practice is a low one, first or second order. This treatment proved successful for glycine13 and is also expected to work well for other moderately anharmonic systems. For strongly anharmonic systems, the anharmonic interactions may be too strong for perturbation-theoretic treatment, and there is a concern that the approach may fail. It seems that so far this has not been attempted for strongly anharmonic biological molecules or complexes. (c) The quantum diffusion monte carlo method The quantum diffusion monte carlo (QDMC) method has been used to compute the zero-point vibrational states of the following very floppy hydrogen-bonded clusters of nucleobases with water molecules:14,15 uracil–water; thymine–water; and cytosine with one and two water molecules. The QDMC algorithm has the merit of being rigorous, that is, it can be converged to ‘‘numerically exact’’ results.Furthermore, the QDMC is also valid for extremely floppy systems, where anharmonic coupling can be very strong. The method is, however, not generally applicable in its standard form to excited states. Applications to vibrational transitions of biological molecules are expected to demand a large computational effort. Nevertheless, this algorithm merits increased attention in the future, in applications of vibrational spectroscopy to biomolecules. (d) Vibrational-self consistent field (VSCF) and correlationcorrected VSCF (CC-VSCF) methods The main method that has been used so far in anharmonic vibrational spectroscopy calculations for biological molecules, small and large, is the vibrational self-consistent field method, including its extensions such as the correlation-corrected VSCF.9 The first versions of VSCF were introduced with a perspective of applications to systems of a few coupled anharmonic oscillators.16–19 Only relatively recently were variants of the method developed that scale as N2, while providing sufficient accuracy.9,10,20 The CC-VSCF method and the direct calculation of spectroscopy from ab initio potential surfaces are described in more detail below.Applications to molecules where an analytic force field is available are computationally very effective, and anharmonic spectroscopy calculations using such potential functions were reported for monosaccharides,21 peptides22 and even a protein.23,24 Such applications may, in cases, be used for providing qualitative insights; the calculation of the fundamental vibrational transitions of the protein BPTI23,24 may, for instance, provide useful predictions for future experiments on characteristic features of protein spectroscopy.The main application of VSCF has, however, been to the testing of available force fields by comparison of the calculated spectra with experimental data. The errors of the VSCF, and in particular, CC-VSCF approximations are sufficiently small enough, that the differences between computed and measured spectra can be attributed mostly to the inaccuracy of the potential function used.9 This is the basis for testing force fields, or for the determination of improved force fields by fitting spectroscopic data with this approach.III. The theoretical basis of the CC-VSCF method An outline of the standard version of CC-VSCF is as follows.9,10,20 As a first step, the molecular geometry corresponding to the relevant local minimum of the potential energy function is computed. The vibrational spectra associated with different local minima (isomers or conformers) of a given molecule are, of course, generally different. Then, the normal mode coordinates corresponding to the minimum are computed by the standard procedure25 (the merits of using normal mode coordinates in VSCF will be mentioned below).The Schrodinger equation in mass-weighted normal mode coordinates25 can be written: { 1 2XN j~1 L2 LQ2j zV(Q1,:::,QN) " #Yn(Q1,:::,QN)~EnYn(Q1,:::,QN) (1) where V is the potential function, and N is the number of normal modes. Eqn. (1) neglects vibration–rotation coupling that in some cases can play a role even for J~0,26 but for large molecules (even the smallest biological molecules) this should not be significant. VSCF, at the simplest level of approximation, makes the ansatz: Y(n)(Q1,:::,QN)~YN j~1 y(n) j (Qj ) (2) The harmonic approximation is, of course, avoided. However, the error due to the separability assumption made, depends on the coordinates used. In situations not very far from the harmonic case, for low-lying vibrational states, normal-mode coordinates provide good approximate separability.9 This leads to the VSCF equations: { 1 2 L2 LQ2j zV (n) j (Qj) " #y(n) j (Qj )~e(n) j y(n) j (Qj ) (3) where the effective potential V (n) j (Qj ) for mode Qj is given by: PhysChemComm, 2002, 5(21), 142–150 143V (n) j (Qj)~SYN l=j y(n) l (Ql ) V(Q1,:::,QN) j jYN l=j y(n) l (Ql )T (4) Eqn. (3) and (4) for the single-mode wavefunctions, energies and effective potentials must be solved self-consistently.A variety of methods for solving the single-mode Schrodinger equation [eqn. (3)] can be employed. Both the ground and excited states of eqn. (3), and therefore the ground and excited VSCF states of the total system can be obtained. The VSCF approximation for the total energy is given by:9 En~XN j~1 e(n) j {(N{1)SYN j~1 y(n) j (Qj ) V(Q1,:::,QN) j jYN j~1 y(n) j (Qj)T (5) As it turns out, the simple VSCF approximation described above is in many cases of insufficient accuracy, e.g., for testing potential functions.Sufficient accuracy can be obtained by improving on VSCF, and including correlation effects between different vibrational modes in the total vibrational wavefunction. Since these correlation effects are usually not very large, they can be obtained by perturbation theory.9,20 The full Hamiltonian is written in the form: H~HSCF,(n)zDV(Q1,:::,QN) (6) where HSCF,(n)~XN j~1 { 1 2 L2 LQ2j zV (n) j (Qj ) " # (7) Applying standard second-order perturbation theory to the Hamiltonian as written in eqn. (6) and (7), with DV as the perturbation, yields the following solution for the energy: ECC-VSCF n ~EVSCF n zXm=n SQN j~1 y(n) j (Qj) DV j jQN k~1 y(m) k (Qk)T 2 E(0) n {E(0) m (8) All the wavefunctions y(m) k and the energies E(0) m in eqn.(8) are computed from the VSCF eqn. (3) and (4). EVSCF n is the total VSCF energy given by (5), and E(0) m ~XN j~1 e(n)m j (9) where e(n)m j is the mth energy level of mode j, computed from the effective potential V (n) j (Qj ), defined in eqn. (4). In most applications so far, the second-order perturbation-theoretic treatment seemed to provide sufficient accuracy. The algorithm is analogous to Moller–Plesset theory in electronic structure calculations, and specifically at this level, to MP2.27,28 Higherorder perturbation theory may prove necessary in cases where increased accuracy is required, probably in cases of small vibrational splittings.29 Near-degeneracy of excited vibrational states may cause problems in CC-VSCF, due to small denominators in eqn.(8). A possible remedy in this case is to treat the effects of correlations between different vibrational modes by degenerate perturbation theory, as demonstrated by Matsunaga et al.30 One may, of course, avoid the problem at the cost of loss of accuracy, by treating the near-degeneracy of states at the level of VSCF alone. Without simplifications of the potential function, the VSCF and CC-VSCF as presented above, are prohibitively expensive computations for large molecules. The difficulty is due to the multidimensional integrals, e.g. eqn. (4), that scale exponentially with N. Fortunately, two alternative approximations were found that hugely decrease the computational effort required in VSCF and CC-VSCF, with an acceptable cost of accuracy, at least for essentially all systems studied so far.9 The first approach is to use a power expansion of the potential in the normal-mode coordinates:9,22-24,31 V(Q1,:::,QN)~ X m1 ,:::,mN Vm1 ,:::,mN (Q1)m1 :::(QN)mN (10) With this, VSCF and CC-VSCF energies can be expressed, of course, with only one-dimensional integrals.However, for many very floppy systems, e.g. (H2O)n clusters, the power expansion either diverges or converges very poorly.20 For the case of potentials not available in explicit, analytic form, (the ab initio potential functions that will be discussed below), highorder terms in the expansion are practically impossible to compute.Even for quartic force fields, PN j~1 mjƒ4 [see eqn. (10), above], which are the minimum required even for relatively easy cases, the calculation of the expansion coefficients for ab initio coefficients is extremely demanding for large molecules. The approach based on eqn. (10) is thus useful mostly for analytical potentials, and for systems that are only moderately anharmonic. The second approximation used in CC-VSCF is to include only interactions between pairs of normal modes.20 Direct coupling terms involving three or more normal modes in the potential are neglected. With this, the only coupling terms between different modes that need to be considered are: Vij (Qi ,Qj)~ V(0,:::,Qi ,:::,Qj ,:::,0){V(0,:::,Qi ,:::,0){V(0,:::,Qj ,:::,0) (11) Note thatV(0,:::,Qi ,:::,Qj ,:::,0) in (11) denotes the potential function computed for Ql ~ 0, for all modes l | i, j.This approximation must, of course, be tested in principle for each case of application. So far, it has proved very satisfactory both for chemically-bound molecules and for highly-anharmonic, weakly-bound clusters.20,32 Combined with the simplification of the potential, the computational effort of CC-VSCF scales with N (in the case of potential expansion) or with N2 at most (in the case of the pairwise-coupling approximation). In either case, if the potential is available analytically, calculations of systems having up to thousands of coupled modes (i.e. proteins) are feasible, for example, see ref. 23 and 24. IV. Direct calculation of anharmonic vibrational spectroscopy from ab initio potential energy surfaces An extremely important methodological issue is the direct calculation of anharmonic vibrational spectroscopy from ab initio potential energy surfaces. This was accomplished via an algorithm that generates potential energy points from electronic structure calculations, and employs these points directly in the CC-VSCF expression.10 The potential must be computed for the multidimensional surface (composed of one- and twodimensional grids of points), which covers the space of nuclear configurations pertinent to the vibrational transitions of interest.For example, in the case of the glycine molecule, which has three low-energy conformers, a total of approximately 50,000 potential energy surface points were necessary.11 These points cover the space of nuclear configurations pertinent to the fundamental transitions associated with all 24 vibrational modes of the system.These calculations, as all ‘‘direct’’ calculations of the anharmonic spectroscopy from ab initio potentials that have been done so far, use the pairwisecoupling assumption for the interactions between the normal modes, as discussed above. So far, such ab initio vibrational spectroscopy calculations, with coupling between all modes of the system, were done only for systems of up to about fifteen atoms.11,33,34 The electronic structure method used in most of the ab initio vibrational spectroscopy calculations so far has been MP2, which seems to provide potential surfaces of very satisfactory 144 PhysChemComm, 2002, 5(21), 142–150accuracy.11,33,34 (For similar calculations of MP2 surfaces, using a different VSCF algorithm, see ref.35.) However, ab initio vibrational spectroscopy calculations were also done with potential energy points from density functional theory (DFT).36,37 We will discuss the merits of different potentials later, but high level energy functionals give results that, by the test of spectroscopy, are competitive with MP2.37 We will see in the next section, that based on comparison with experimental data for systems such as glycine11 or N-methylacetamide,34 the ab initio potentials seem considerably superior to the standard analytical-empirical force fields in use, such as AMBER38 or OPLS.39 It is very useful that ab initio vibrational spectroscopy with VSCF and CC-VSCF has now been incorporated into the electronic structure code package GAMESS,40 for the convenience of potentially interested users.VSCF and CC-VSCF were also very recently applied to spectroscopy calculations for hybrid ab initio/empirical potentials 33 and for potentials from semiempirical electronic structure methods.41 Both of these approaches also make calculations possible for large biological molecules. The preliminary results obtained, will be discussed later. V. Vibrational spectroscopy and force fields of small biological molecules The recent calculations of vibrational spectroscopy of small biological molecules and the comparison of the results with experiment shed light on the potential energy surfaces of the molecules, including the anharmonic interactions.Nearly all the analysis and discussion below is based on CC-VSCF calculations for a few systems that serve for our purposes as benchmark cases, including glycine, N-methylacetamide and glycine–water. (a) Comparison of theory with experimental results It is useful to examine first, the differences between computed and measured spectroscopy, such as those that can be obtained with the most accurate potential surfaces available at present. The comparison for glycine is quite informative. Glycine has three conformers of low energy, according to MP2/DZP calculations;11 conformer 2 is only 0.63 kcal mol21 in total energy above conformer 1, and conformer 3 is 1.40 kcal mol21 above the lowest-energy structure.The three conformers, shown in Fig. 1, are believed to be present in matrix-isolation experiments that were carried out,1b and to contribute to the IR spectroscopy. Detailed comparisons between experimental results and CC-VSCF calculations using an ab initio MP2 potential surface are found in ref. 11. Fig. 2 shows the comparison between computed and experimental infrared spectroscopy frequencies for the lowest energy conformer of glycine. (Only those modes whose transition dipole is sufficiently strong are observed experimentally.) Mode 1 is the highest-frequency mode, and modes were enumerated in decreasing order of the harmonic frequencies. All of the transitions presented are fundamentals. Fig. 3 shows the percentage deviation between theory and experiment for the transitions that were observed. The deviations for OH stretching vibration (mode 1), NH asymmetric stretching vibration (mode 2) and CH symmetric stretch (mode 5), are in the range of about 0.1 to 1.0 percent.For the HNH and HCH bends (modes 7 and 8) the percentage deviation is about 2.5-3.0 percent. Even for soft skeletal vibrations, the deviations are quite small, less than 3% for mode 20, with a frequency of 500 cm21, and practically zero for mode 21, with a frequency of 463 cm21. With such small deviations, the comparison provides strong support for the ab initio potential used. The same conclusion can roughly be drawn from the results for conformers 2 and 3, with the main exception that the deviation for the only soft vibration measured in this case (mode 21) is around 8%.The Fig. 1 The structure of the three lowest energy conformers of glycine. Fig. 2 Comparison of measured and calculated fundamental vibrational frequencies of the lowest energy conformer of glycine. The mode number corresponds to the mode excited in the transition. Calculations are from ref. 11 and experimental data is from ref. 1b. Frequencies are in cm21. PhysChemComm, 2002, 5(21), 142–150 145experimental errors may, perhaps, play some role, since the effect of the rare gas matrix, estimated to be minor, is not yet known with confidence. Interestingly, the few transitions measured in superfluid He droplets are in even better agreement with theory than the matrix experiments,4 but the data available are insufficient for a firm conclusion on this.Analysis of the results for N-methylacetamide34 (structure given in Fig. 4), a molecule that can serve as a model for the peptide bond and its force field, leads to quite similar conclusions. Comparison of theoretical and experimental frequencies is shown in Fig. 5, and the percentage deviations of the calculations from the (matrix) experiments for the different fundamental transitions (not all observed) are given in Fig. 6. The CC-VSCF calculations were also in this case carried out for ab initio MP2 potential surfaces.34 For the hydrogenic stretching modes; the NH stretching vibration (mode 1); the CCH3 asymmetric stretches (modes 2 and 4); the NH3 asymmetric stretches (modes 3 and 5); the NCH3 symmetric stretch (mode 6); and the CCH3 symmetric stretch (mode 7), there is excellent agreement between the calculated and experimental results, with deviations ranging from 0.05 to 0.7%. For the transitions associated with the amide group, amide I (mode 8), amide II (mode 9) and amide III (mode 16), the agreement is also very good, though the percentage deviations (less than 2.5%) are higher than for the hydrogenic stretches.Given the recognized difficulty of correctly predicting low-frequency transitions, the results are also very good, with the exception of the in-plane-bend of the CO with respect to the CCH3 group (mode 26), in which case we assume that the ab initio potential does not describe this mode accurately enough. The most disappointing comparisons are, on the whole, for various bending motions associated with the NCH3 and CCH3 Fig.3 Percentage deviation between calculated and measured fundamental frequencies of the lowest-energy conformer of glycine. The mode number corresponds to the mode excited in the transition. Fig. 4 Structure of N-methylacetamide. Fig. 5 Comparison of measured and calculated fundamental frequencies of trans-N-methylacetamide. Calculations are from ref. 34 and experimental data is from ref. 1a. Frequencies are in cm21. Fig. 6 Percentage deviations between calculated and measured fundamental frequencies of trans-N-methylacetamide. 146 PhysChemComm, 2002, 5(21), 142–150groups (modes 10 through 15), where the deviations are above 5%, and in our view must be due to the limitations of the ab initio force field. From the examples of the amino acid and the peptide model, we conclude that CC-VSCF anharmonic spectroscopic calculations, using ab initio potentials, give an accurate description of most fundamental transitions, which in turn supports the validity of the MP2 potential surfaces.(b) The role of anharmonic effects Having established the good accuracy of MP2 potential surfaces, we now examine the contributions of anharmonic effects to the fundamental frequencies for these force fields. Fig. 7 shows Dvn/vn versus n, where vn is the fundamental frequency of mode n, and Dvn is the computed difference between the anharmonic (CC-VSCF) and harmonic frequencies, for the lowest-energy conformer of glycine. The results are from ref. 11. For the hydrogenic stretching vibrations, OH (mode 1), NH2 (asymmetric and symmetric, modes 2 and 3 , respectively), and CH3 (asymmetric and symmetric, modes 4 and 5 respectively), the anharmonic effects are quite large, and the contribution of anharmonicity to the observable transition frequency can exceed 5%.For the stiff bending modes, such as HNH (mode 7) and HCH (mode 8), the anharmonicity is, on the other hand much smaller, its relative contribution to the fundamental frequency being on the order of 2%. For the softest, skeletal modes the role of the anharmonic contributions seems to vary broadly. For some low frequency collective vibrations, e.g. mode 22, the effect of anharmonicity can be quite moderate. For soft modes with partial torsional character, such as mode 23 turns out to be, the effect of anharmonicity can be extremely large, with the anharmonic effects making a contribution on the same order of magnitude as the harmonic effects.Collective modes of high anharmonicity are of considerable interest. From the pragmatic point of view pursued here, we draw attention to the fact that for such modes pure harmonic calculations are essentially futile. Careful studies of such transitions could provide useful information about the potential energy surfaces. Unfortunately, very often experimental information on these transitions is unavailable (as in the present case), since the intensities are not strong. As for all cases of low frequencies, the accuracy of the potential functions with regard to the pertinent force constants is questionable. In summary, a combined experimental and theoretical effort is called for in the study of low-frequency collective modes. This is likely to provide important improvements in this respect, of available force fields, which are subject to considerable uncertainty at present.On the other hand, progress can readily be expected for the hydrogenic stretching modes; the anharmonic effects are fairly large, data is available, and the potentials at hand seem accurate. The use of anharmonic calculations for such modes is both desirable and straightforward. Fig. 8 shows Dvn/vn, the relative anharmonic contribution to the transition frequency, versus n, for N-methylacetamide.34 In general, the results follow the same pattern as for glycine; for the hydrogenic stretching modes, the anharmonic effects are relatively large, for the stiff bending modes they are very moderate, and for some of the very soft collective modes they can be extremely large.Systems for which some very large anharmonic effects are expected, are hydrogen-bonded complexes of biological molecules, e.g. their complexes with water. The 1 : 1 complex of glycine with water may be a useful representative case. The structure of the most stable isomer of glycine–water is shown in Fig. 9.33 In Fig. 10, we show the relative contribution of the anharmonicity to the transition frequency Dvn/vn, for several fundamental transitions, n.We listed only the seven hydrogenic stretching modes, and then, the eight lowest frequency modes. For the X–H stretching vibrations, whether pertaining to the glycine or to the water component, the behavior is similar to that discussed previously for glycine and N-methylacetamide. For the soft modes, the anharmonic effects are extremely large, some of them correspond primarily to soft collective modes of glycine (modes 22, 29 and 33).The other modes listed are Fig. 7 Dvn/vn (percentage) versus the mode number n for the fundamental transitions of the lowest-energy isomer of glycine. Dvn is the anharmonic contribution to the transition frequency vn. The calculations are from ref. 11. Fig. 8 Dvn/vn (percentage) versus the mode number n for the fundamental transitions of trans-N-methylacetamide. Dvn is the anharmonic contribution to the transition frequency vn. The calculations are from ref. 34. PhysChemComm, 2002, 5(21), 142–150 147intermolecular in nature; mode 30 corresponds to the OH(gly)– O(H2O) stretch, and mode 31 to the CLO(gly)–O(H2O) stretch.Interestingly, in these two cases the anharmonic effects are not extremely large; for mode 31 they are very substantial, but only on the order of 11%. For mode 30, the effects are very modest, which we attribute partly to geometric considerations (the ring structure acts to reduce the anharmonicity), and partly to the cancellation of different types of anharmonic effects (the intrinsic anharmonicity of the modes versus the coupling contributions with other modes). For several ‘‘intermolecular’’ soft modes (modes 32 and 26), the anharmonic effects are extremely large, though glycine–water is vibrationally stiffer than clusters such as (H2O)2.Spectroscopy is thus sensitive to anharmonic effects, and can be used to test the anharmonic interactions in a given potential. The effects on transition frequencies are in most cases not extremely large, since, clearly the harmonic contribution nearly always dominates. The anharmonic interactions will, of course, govern vibrational energy transfer between different modes. Within the harmonic approximation, there is certainly no energy transfer between different normal modes. Recently, there has been considerable interest in vibrational energy flow in large biological molecules.42,43 Anharmonic couplings extracted from spectroscopy may help quantify the models that have been proposed for this vibrational energy flow. (c) Merits and disadvantages of different types of force fields (i) MP2 Potential energy surfaces. So far, the results we have discussed were for ab initio force fields, specifically for MP2 potential energy surfaces.The spectroscopic calculations use energy values computed on a large multidimensional surface (composed of one- and two-dimensional grids of points) that must cover the range of nuclear configurations pertinent to the spectroscopic transitions of interest. It is not surprising that MP2, used in the various applications with a DZP basis set,11,33,34 yields very good results. This level of electronic structure theory has been extensively used in calculations of conformer structures of small biological molecules, and there is evidence that it performs very well.44,45 Although MP2, is on the whole very satisfactory, it is useful to recognize that it is not completely satisfactory for all purposes; as pointed out in subsection V-(a), it gives excellent results for hydrogenic stretching vibrations, for instance, but it can give significant errors for the vibrations associated with low-frequency collective modes.Additionally, force constants for certain distortion or bending modes of higher frequencies can have appreciable errors, according to the test of vibrational spectroscopy. The main limitation of MP2 is, however, that of computational cost. Given the need to evaluate a large grid of points in order to represent the force field for anharmonic spectroscopy calculations, systems of 15–20 atoms (with all modes included) probably represent the largest feasible target at present, or in the near future.(ii) Coupled cluster methods. Higher-level electronic structure methods,46 e.g. coupled cluster methods, are probably out of the question for use in CC-VSCF calculations of biological molecules. The vibrational spectroscopy seems sensitive to the shape of the potential energy surface. Based on our experience, it seems better to remain within one consistent method such as MP2, than to try and calculate, for example, only the harmonic part by a higher level method. (iii) Density functional theory. As discussed in Section IV, in addition to MP2, other electronic structure methods were used with CC-VSCF for ab initio vibrational spectroscopy. In particular, density functional theory (DFT) with several energydensity functionals, was tested for smaller, less demanding, systems such as formic and acetic acids.36,37 The comparisons with experiment make it possible to rank the different electronic structure methods according to the spectroscopic quality of the potential surfaces.Based on the results available so far, a very tentative ranking is: B97 ¢ MP2 ¢ B3LYP w HCTH w BLYP All of the methods listed above, except for MP2, utilize different DFT functionals. It is clear from the above order that hybrid functionals,47 i.e. B3LYP48 and B9749 give better results than the generalized gradient approximation (GGA) functionals 47 such as BLYP50 and HCTH.51 Within the GGA functionals, the modern HCTH appears to give very good Fig. 9 The structure of the lowest energy isomer of glycine–water. Fig.10 Dvn/vn (percentage) for several modes versus the mode number n for the fundamental transitions of the lowest-energy isomer of glycine–water. Dvn is the anharmonic contribution to the computed transition frequency, vn. (The calculations are from ref. 33.) 148 PhysChemComm, 2002, 5(21), 142–150results, well superior to BLYP. A more rigorous classification of the electronic structure methods with regard to their spectroscopic performance should be quite useful. For molecules of the sizes studied here, the computational effort in using B97 and MP2 is about the same. (iv) Empirical force fields. The most extensively used force fields in the modeling of biological molecules are clearly the empirically-based analytical potential functions such as AMBER, OPLS-AA and others.52 Comparisons with spectroscopic data for glycine,11 N-methylacetamide34 and several other molecules have shown that the standard empirical force field methods tested are much less accurate than the ab initio potentials.11,34 This is not surprising, since these methods were not calibrated for spectroscopy.Nevertheless, one learns from these comparisons about the extent and nature of the shortcomings of the available empirical force fields. For example, in the standard versions of AMBER and OPLS, the interactions between adjacent chemically-bound atoms are modeled as harmonic with respect to the interatomic distance. There are sources of anharmonicity, of course, in these force fields: Coulomb interactions between partial charges on certain atoms; torsional potential terms; van der Waals interactions; etc.However, these force fields seem inadequate, at least, for spectroscopy. An important weakness of the empirical force fields is in the modeling of hydrogen-bonded interactions; comparisons with ab initio calculations for glycine–H2O show huge differences between the OPLS potential surfaces, and the ab initio one.33 It is undoubtedly possible, in principle, to develop much improved empirical, analytical force fields, and important work of this type is underway in several groups. At the present state of the art, ab initio potentials are much more preferable for systems for which their calculation is feasible. (v) QM/MM. There are some preliminary results that may affect the development of improved force fields for much larger systems.The first of these is the application of hybrid potential surfaces that treat part of the molecule by ab initio calculations, and the remainder by empirical force fields. Such methods, generally referred to as QM/MM,53-56 are a very active direction of research for molecular modeling. By treating a ‘‘problematic’’ small region of the molecule by an ab initio method, such an approach can yield much better spectroscopic results than the standard force fields, as shown in a recent study of the glycine–water complex.33 (vi) Semi-empirical methods. Another promising direction may be found in the use of potential surfaces based on semiempirical electronic structure methods.41 In attempts to apply CC-VSCF to potential surfaces from semi-empirical electronic structure algorithms, e.g.PM3, in their standard forms, the spectroscopic results had percentage deviations that varied between 0.5 to more than 20%.41 However, when these potentials were modified (via a scaling procedure) by using information from ab initio calculations, improved and very encouraging results were obtained. Both QM/MM and the ‘‘scaled’’ semi-empirical electronic structure methods are, however, at early stages of investigation for the purpose of vibrational spectroscopy. VI. Future directions As described in Section V, results show that anharmonic vibrational spectroscopy calculations are already a practical tool for quantitative interpretation of experiments and insights into the quantitative properties of the potential energy surfaces of small biological molecules.There are many possible applications to specific areas of great interest, in particular, sorting out issues of spectroscopic assignment and relating them to features of the potential surfaces. There are, however, several extensions in new directions that have hardly been pursued, as yet. One very important possible direction of future application is the spectroscopy of excited potential energy surfaces. Recent experiments, e.g. from the groups of Simmons,2 de Vries,3 Kleinermanns3 and several others are providing, in cases, vibrational information about the excited electronic states of biological molecules. Present knowledge about the excited state potential surfaces is extremely limited, much more so than for the electronic ground state.A second major challenge is the study of larger molecules, where the tool of ab initio calculations, so successful for small molecules, is not applicable. We mentioned the possibilities of QM/MM hybrid potentials33 and also of improved semiempirical electronic structure methods41 as tools for the spectroscopy of larger systems. It is also possible that new, improved empirical force fields can be developed by fitting procedures. Finally, the issue of developing potential functions that are suitable for application both to spectroscopy and in calculation of other properties, deserves attention. The ab initio methods, which are most useful for spectroscopy, provide points on the potential energy surfaces, which can be used directly in the quantum-mechanical spectroscopy calculations (CC-VSCF). 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ISSN:1460-2733
DOI:10.1039/b208000a
出版商:RSC
年代:2002
数据来源: RSC