J. Phys. Chem. C 2010, 114, 20603–20608
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On the Mean Accuracy of the Separable VSCF Approximation for Large Molecules† Liat Pele‡ and R. Benny Gerber*,‡,§ Institute of Chemistry and The Fritz Haber Research Center, The Hebrew UniVersity of Jerusalem, Jerusalem 91904, Israel, and Department of Chemistry, UniVersity of California IrVine, IrVine, California 92697 ReceiVed: June 2, 2010; ReVised Manuscript ReceiVed: July 26, 2010
The separable VSCF approximation and the VSCF-PT2 method are extensively used for anharmonic vibrational spectroscopy calculations of large molecules. VSCF-PT2 uses second-order perturbation theory to correct the VSCF level, and is thus more accurate. It is shown by test calculations for a series of amino acids and peptides that the mean deviation of VSCF from VSCF-PT2 frequencies decreases for an increasing number of modes N, the decrease scaling roughly with log N. It is conjectured that the result is a manifestation of improved mean accuracy of VSCF as a mean-field approximation, a consequence of increased averaging with increasing numbers of modes. There is no systematic increase in VSCF accuracy with N for individual vibrational transitions. Increased accuracy of VSCF with N is found for certain groups of transitions, e.g., N-H stretches. The results are expected to be useful in choosing methods for spectroscopy calculations of extended systems such as large peptides, proteins, and nucleic acids. I. Introduction Vibrational spectroscopy experiments are among the most powerful and widely used tools for exploring properties of molecular systems. The usefulness of spectroscopy experiments for extracting information on properties, such as the potential energy surfaces of the studied molecules, depends on the availability of suitable theoretical tools of interpretation. In recent years, there have been major advances in spectroscopic techniques with much improved resolution for large molecules.1-8 Studies of biological molecules in beams are one area where great progress has been made.3-8 Such experiments pose a major challenge to the computational methods for vibrational spectroscopy, since good accuracy for large systems is required. The present paper deals with computational methods beyond the harmonic approximation, since the latter is known to be inadequate in many applications.9-15 Empirical scaling of the harmonic frequencies is often used16-19 and provides improved agreement with experiments. However, this is not a first-principles approach, and therefore fails to provide insights on the properties of the underlying potential energy surfaces. In the present paper, we explore the relative accuracy of two methods for anharmonic calculations of vibrational spectroscopy, focusing on the behavior for a large number of degrees of freedom N. One of the methods, the vibrational self-consistent field (VSCF) approximation, assumes separability of the full vibrational wave function in the different vibrational degrees of freedom.20-23 The VSCF approximation was proposed years ago by Bowman,20 Gerber and Ratner, 21 and other authors, and its effectiveness in applications has been enhanced in a variety of ways introduced to represent the potential surface, and by employing normal modes for the vibrational degrees of freedom.24-35 Recently Hansen et al. presented a new implementation of VSCF that is accelerated by an order of magnitude.36 VSCF †
Part of the “Mark A. Ratner Festschrift”. * To whom correspondence should be addressed. ‡ The Hebrew University of Jerusalem. § University of California Irvine.
with its various improvements has emerged as a computationally efficient tool for anharmonic spectroscopy calculations, with applications that include small biological molecules,14,27,37,38 peptides,39 biomolecules in crystals,40-42 and even the protein BPTI.13 However, the separability approximation significantly limits the accuracy of VSCF, though this method is generally more accurate than the harmonic approximation. Several methods have been introduced that extend beyond the separability approximation, and offer better accuracy than the basic VSCF.43-48 A method that corrects the VSCF energies, keeping the computational effectiveness for relatively large molecules, is VSCF-PT2, also referred to in the literature as CC-VSCF, and as vibrational Møller-Plesset perturbation theory (VMP).49 This approach, ascribed to Jung and Gerber25 and Norris et al.,49 treats the corrections for nonseparability for VSCF by secondorder perturbation theory. VSCF-PT2, as we dub the method here, is generally more accurate than VSCF, working within the approximation af separability, but computationally more demanding. In recent years, the method has been applied to a range of biological molecules and their complexes, yielding onthewholeverysatisfactoryagreementwithexperiments.14,15,25,42,50-53 Algorithms have been introduced that improve the scaling of VSCF-PT2 with N, the number of degrees of freedom.47,54 However, standard VSCF is computationally more efficient, and for very large systems, e.g., proteins, the difference is important. The main objective of this paper is to show that the mean deviation of VSCF from VSCF-PT2 decreases asymptotically for very large N, the decrease scaling roughly with log N. It is conjectured that this is due to the mean-field property of standard VSCF, implying that the mean accuracy of the separability approximation actually improves asymptotically for large N, as averaging is more extensive. The paper is organized as follows: The VSCF and VSCFPT2 approximations and the test systems are briefly described in section II. The results and their analyses are the topic of section III. Concluding remarks are presented in section IV.
10.1021/jp105066a 2010 American Chemical Society Published on Web 08/13/2010
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Pele and Gerber
II. Methods and Systems a. Description of the VSCF and VSCF-PT2 Approximations. We consider a system with negligible rotational-vibrational coupling effects, near a minimum energy configuration. The full vibrational Schro¨dinger equation is
[
-
j
∑ Vjdiag(Qj) + ∑ ∑ Wijcoup(Qi,Qj) i
j>i
(6)
1 ∂2 + V(Q1,...,QN) Ψn(Q1,...,QN) ) 2 j)1 ∂Q2
∑
N
V(Q1,...,QN) )
j)1
]
N
In this study, we employ the following pairwise representation of the potential:25
ΕnΨn(Q1,...,QN)
(1)
This equation is in mass-weighted normal coordinates Q1, Q2, ..., QN, where V is the potential energy function of the system, n is the state quantum number, and N is the number of vibrational modes. Solving eq 1 exactly is not computationally feasible for polyatomic systems. In this study we deal with three approximations for this problem: 1. The Harmonic Approximation. The harmonic approximation involves the calculation and diagonalization of the second derivative matrix with respect to the nuclear displacement, which yields the normal mode coordinates, analytical vibrational wave functions, and the harmonic frequencies. Within to the harmonic approximation, the wave function in normal mode coordinates is separable:
In this approximation, the potential is written as a sum of terms that include single-mode potentials and interactions between pairs of normal modes. The contributions to the potential from higher order interactions of normal modes are neglected. This approximate representation has proved successful in large numbers of applications, including for biological molecules.15,50,51 3. The VSCF-PT2 Approximation. The results of the VSCF approximation can be improved by using second-order perturbation theory; this method is referred to as VSCF-PT2:
H ) HSCF,(n) + ∆V(Q1,...,QN)
where HSCF,(n) is the Hamiltonian in the VSCF approximation, and ∆V(Q1, ..., QN) is treated as a small perturbation. Applying second-order perturbation theory for the energy levels yields N
|〈
N
Ψn(Q1,...,QN) )
∏ ψj(n)(Qj)
(2)
j)1
(7)
ΕPT2 ) Ε(0) n n +
∑
∏
N
ψj(n)(Qj)|∆V|
j)1
Ε(0) n
m*n
-
∏ ψj(m)(Qj)〉|2
j)1 Εm(0)
(8) 2. The VSCF Approximation. In the standard version of VSCF approximation, the normal mode coordinates are employed. The separability of the wave function (eq 2) is taken to be ansatz. This leads to the single-mode equation:
[
-
]
1 ∂2 ¯ j(n)(Qj) ψj(n)(Qj) ) εj(n)ψj(n)(Qj) +V 2 ∂Q2 j
(3)
The effective potential Vj(n) j (Qj) for the mode Qj is given by a mean field approximation: N
N
l*j
l*j
j jn(Qj) ) 〈 ∏ ψl(n)(Ql)|V(Q1,...,QN)| ∏ ψl(n)(Ql)〉 V
(4)
Equations 3 and 4 are solved self-consistently.
where EnPT2is the correlation-corrected energy of state n, and En(0) are the VSCF energies. Note that VSCF-PT2 ignores the degeneracy effect. These can be addressed in a VSCF methodologybutrequirecomputationallymoredemandingcalculations.45,55 b. The Test Systems. Calculations were carried out for mono-, di-, tri-, and tetrapeptides; the main results obtained for peptides used in our computations, together with the number of their vibrational modes, are listed in Table 1. Our assumption is that the number of modes of these systems suffices for a meaningful statistical trend. The spectroscopy calculations were carried out in each of the cases for a single conformer. A search was made for each lowest energy structure of each peptide. The energy minimization search as well as the vibrational calculations were carried out using the GAMESS quantum chemistry package.56 Semiempirical PM357 potentials were used for all the systems. These force fields are computationally efficient and
TABLE 1: Tested Amino Acids and Peptides, the Number of Normal Modes, and Statistics of the VSCF versus VSCF-PT2 Deviation
molecule
# normal modes
mean deviation (%)
standard deviation from mean (%)
the 90th percentile (%)
max deviation (%)
absolute max deviation in cm-1
monoglycine monoalanine diglycine monovaline dialanine triglycine tetraglycine trialanine divaline Val-Gly-Val
24 33 45 51 63 66 87 93 99 120
4.46 4.00 3.40 1.32 1.49 2.27 1.50 1.17 1.73 1.04
9.91 10.34 10.85 2.44 2.45 8.50 3.90 3.08 6.28 3.56
23.05 4.36 5.19 4.03 4.45 4.08 2.91 2.24 3.28 3.46
39.37 51.43 69.95 13.26 12.18 68.6 31.88 27.44 59.96 37.83
131.81 112.14 161.64 88.46 71.28 162.06 106.92 95.82 156.78 113.15
Mean Accuracy of the Separable VSCF Approximation
J. Phys. Chem. C, Vol. 114, No. 48, 2010 20605 spectrum is calculated by three methods: the standard harmonic approximation, the VSCF method, and the VSCF-PT2 method. The accuracy of the harmonic approximation and the VSCF method are compared to the results of VSCF-PT2, under the assumption that it is the most accurate method. Experience with a number of applications using reliable potentials suggests that VSCF-PT2 is more accurate than VSCF in the vast majority of applications, including for biological molecules.14,15,25,42,50-53 To estimate the relative accuracies of the frequency calculations, we averaged the differences between the results of the VSCF-PT2 methods (noted as ωPT2 j ) and the results as calculated ) or the harmonic approximation (noted by VSCF (noted as ωVSCF j as ωhrm j ). The final estimation is given as an average over all transitions, according to the following formulas:
Figure 1. The mean deviation between the VSCF-PT2 and VSCF frequencies as a function of the number of modes. Averaging over all the fundamentals.
∑ |ωjPT2-ωjVSCF|/ωjPT2
(11)
∑ |ωjPT2-ωjhrm|/ωjPT2
(12)
γPT2 )
100 N
γhrm )
100 N
In this study, we focused on the fundamental transitions. Since we include all fundamentals, all parts of the spectrum are given, with equal weight. To examine the effect of different specific types of normal modes, the relative accuracies of the computed frequencies were calculated for H-O and N-H stretches and for C-H stretches. Different trends were found for these specific groups.
Figure 2. The distribution of the deviation between the frequencies of the VSCF-PT2 and the VSCF (in percentages) of Val-Gly-Val.
are not spectroscopically very accurate; i.e., their accuracy was inferior to ab intio methods, such as MP2. However, the potentials are at least realistic.15,58 The accuracy of the potentials used should not affect our considerations on the vibrational algorithms. c. Accuracy Measures. Calculations are carried out for mono-, di-, tri-, and tetrapeptides. For each peptide, the IR
III. Results and Discussion a. The Relative Accuracy of the VSCF Approximation. The mean deviations are shown in Figure 1. The graph depicts the mean deviation between the frequencies of VSCF-PT2 and VSCF as a function of the number of modes N. The graph shows a clear trend of decreasing deviation between the results received by two methods. This decrease can be described by the following regression line: γvscf ) -2.1615 log(N) + 11.128. The percent of the general model’s variance explained by the linear regression model (R2) equals 0.783. The regression was found to be statistically significant (R ) 0.001). We suggest that this can be explained by the nature of the VSCF approximation. The VSCF method involves a mean field approximation averaging of the mutual interactions of normal modes (see eq 4). As the size of the molecule grows, the number of interactions between normal modes grows as well, and more terms are averaged. As more terms are averaged, the results converge to the accurate value. We consider the question of whether the shrinking derivation with N between VSCF and VSCF-PT2 may be due to a decrease
TABLE 2: Tested Amino Acids and Peptides, the Number of Normal Modes, and Statistics of the Harmonic versus VSCF-PT2 Deviation molecule
# normal modes
mean deviation (%)
standard deviation from mean (%)
the 90th percentile(%)
max deviation (%)
absolute max deviation in cm-1
monoglycine monoalanine diglycine monovaline dialanine triglycine tetraglycine trialanine divaline Val-Gly-Val
24 33 45 51 63 66 87 93 99 120
6.43 5.35 7.94 8.60 9.00 10.55 9.70 10.37 9.60 10.27
10.74 7.21 15.02 13.86 15.59 19.67 18.54 17.10 15.48 16.99
16.26 10.08 21.79 35.24 35.34 29.69 38.10 45.20 38.44 42.58
52.07 37.82 74.12 57.46 76.04 85.49 81.00 80.08 78.64 80.79
47.98 42.74 67.17 208.72 190.89 90.27 135.08 171.11 161.44 178.9
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of VSCF-PT2 accuracy. We think this is very unlikely to be the case. First, the available comparisons of VSCF-PT2 with experimental data, however nonsystematic with regard to the types of molecules studied, and however restricted to molecules smaller than those studied here, give no indication of deterioration of the accuracy of VSCF-PT2 (certainly not compared to VSCF) with the size of the system. Second, the derivation of VSCF-PT2 and the structure of equations give no reason to expect a decrease of accuracy with N. This is in contrast with the mean-field argument given above, for the improved accuracy of VSCF. Another point to consider is the following: the variance of the deviation, as shown in Table 1, can not be neglected. Can we certainly claim that the means of the deviations is significantly deferent and not due to arbitrary fluctuations? For this purpose, a one-way ANOVA (analysis of variance)59 was employed. This is a statistical test that is based on the relation between variances between groups and within groups. This test is used to decide whether different populations have significantly different mean values. In this case, each molecule yields a group of deviations between VSCF-MP2 and VSCF, and these groups are found to have different mean values with significance value R ) 0.056. Another test for a linear trend in the one-way ANOVA gave a significance of R ) 0.002 for the derivations from linearity. This implies that the decreasing trend of the mean deviation between the frequencies of VSCF-PT2 and VSCF is not due to arbitrary fluctuations. In cases where one is interested in one or in a few specific transitions, the concerns are different.28,31 The distribution of the deviation is as follows: Most deviations are small. However, a very small fraction of modes have a very large deviation. These modes are mostly low-frequency (primarily less than 600 cm -1). This is illustrated in Figure 2, which depicts the deviation distribution for Val-Gay-Val. Table 1 lists the mean distribution statistics for each molecule. In particular, the table shows that 90% of the deviations are relatively low, at list for the larger molecules. It is also shown that the standard deviation of the deviation does not increase with the size of the molecules; however, the maximum deviation may be considerably larger and may not correlate with the size of the molecule. In cases where only a few transitions are of interest, especially if they are of low frequency, the perturbation correction should not be discarded. Needless to say, here we only compared VSCF with VSCF-PT2, which is also an approximation. However, as pointed out earlier, there is strong evidence that VSCF-PT2 is generally more accurate than VSCF. b. Relative Accuracy of the Harmonic Approximation. The quality of the harmonic approximation with respect to the VSCF-PT2 is presented in Table 2. A one-way (molecules × deviations) ANOVA59 test for differences between the mean deviations did not reach significance. A test for a linear trend in the one-way ANOVA was also nonsignificant with R ) 0.11. These statistical tests imply that the trend shown in Figure 3 may be due to arbitrary fluctuations. This is because the variance of the deviation distribution is very large. The distribution of the deviations is different from the previous case. Here there are more either small or medium-sized deviations. The distribution of the deviations is illustrated for Val-Gly-Val in Figure 4. Table 2 lists the mean statistics for the distributions of each molecule. The table shows that the 90th percentile of the deviations as well as the standard deviation is relatively high. In summary, there is no conclusive evidence from the results as to a trend on the accuracy of the harmonic approximation with N for the systems studied.
Pele and Gerber
Figure 3. The mean deviation between the frequencies of the VSCFPT2 and the harmonic approximation as a function of the number of modes. Averaging over all the fundamentals.
Figure 4. The distribution of the deviation between the frequencies of the VSCF-PT2 and harmonic approximation (in percentages) of ValGly-Val.
c. Specific Subsets of Normal Modes. Figure 5 shows the mean deviation of VSCF from VSCF-PT2 as compared to the number of modes N, for a set of specific transitions, namely, the OH and NH stretches. The deviation of VSCF for this specific class of modes decreases with the number of modes N. However, this result must be viewed as specific to this class of modes. When given a set of modes, the same behavior cannot be expected for all modes, since the statistics of the mean for the given set is less extensive than for all modes. Apparently the number of OH and NH stretches in the example sufficed for a systematic decreasing trend. The result is useful as a prediction. Stretching mode spectroscopy is important in its own right, and there are many experiments that focus on these transitions. To simply show that behavior can be different for another specific class of modes, consider Figure 6 for the C-H stretches. The mean deviation for the C-H stretches appears to fluctuate widely with N, without a clear systematic trend. It seems likely that the number of CH modes for these amino acids and peptides is insufficient to observe systematic behavior of the mean deviation over this the set of modes. Other explanations for the different behavior of OH and of CH modes are,
Mean Accuracy of the Separable VSCF Approximation
Figure 5. The mean deviation between the frequencies of the VSCFPT2 and VSCF as a function of the number of modes. Averaging over OH and NH stretches.
J. Phys. Chem. C, Vol. 114, No. 48, 2010 20607 increasing N. We suggest that this property could be due to the mean field nature of the VSCF approximation. The accuracy of the mean-field approximation is expected to improve with the number of degrees of freedom, since the latter corresponds to more extensive averaging. This implies that the improved mean accuracy with N should be independent of the class of systems studied, the type of potential surfaces employed, and so forth. At the same time, the evidence presented is for peptides only, and it is desirable to extend the study to other classes of molecules. VSCF calculations are generally computationally less demanding than VSCF-PT2. Thus, the result encourages the use of VSCF for large peptides, proteins and other extended molecules, at least for cases where the mean accuracy, as opposed to the accuracy of a specific transition or a group of transitions, is crucial. This should significantly reduce the challenge of anharmonic spectroscopy calculations for extended biomolecules. Acknowledgment. This paper is dedicated to Mark Ratner. R.B.G. recalls with pleasure many years of fruitful cooperation with M.A.R. We thank Dr.Yair Goldberg for very helpful statistical suggestions. We thank Dr. Dorit Shemesh, Ehud Tsivion, and Jirˇ´ı Sˇebek for revision of the paper and for helpful comments. Supporting Information Available: Table of all the deviations between the VSCF-PT2 and VSCF as well as between VSCF-PT2 and the harmonic. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes
Figure 6. The mean deviation between the frequencies of the VSCFPT2 and VSCF as a function of the number of modes. Averaging over CH stretches.
however, possible. The different number of couplings between OH and CH, respectively, and other modes may also play a role. d. Computational Implications. The computational effort of VSCF and of VSCF-PT2 depends on the complexity of the potential representation. In this study we employed the pairwise representation of the potential (eq 6). In that case, the CPU time for VSCF is better by a factor of 2. For higher levels of potential representation (e.g., if triples coupling terms are included), the computational effort of VSCF-PT2 is larger by orders of magnitude then that of VSCF, and therefore VSCF should be much faster for calculations. IV. Concluding Remarks The main result of this paper, obtained by numerical “experiments” on a series of amino acids and peptides, is that the mean deviation of VSCF frequencies from VSCF-PT2 frequencies decreases with the number of modes N. The decrease scales as log N for sufficiently large N. Since VSCF-PT2, both theoretically and empirically, is more accurate than VSCF, it seems reasonable to assume that the mean accuracy of VSCF (compared with the unknown true results) improves with
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JP105066A
