# Fast and accurate hybrid QM//MM approach for computing anharmonic corrections to vibrational frequencies

**Part of the following topical collections:**

## Abstract

We have developed and tested a new time-effective and accurate hybrid QM//MM generalized second-order vibrational perturbation theory (GVPT2) approach. In this approach, two different levels of theory were used, a high level one (DFT) for computing the harmonic spectrum and a lower fast one (Molecular Mechanic) for the anharmonic corrections. To validate our approach, we used B2PLYP/def2-TZVPP as the high-level method, and the MMFF94 method for the anharmonic corrections as the low-level method. The calculations were carried out on 28 molecules (containing from 2 to 47 atoms) covering a broad range of vibrational modes present in organic molecules. We find that this fast hybrid method reproduces the experimental frequencies with a very good accuracy for organic and bio-molecules. The root-mean-square deviation (RMSD) is about 27 cm ^{-1} while the full B3LYP/SNSD simulation reproduces the experimental values with a RMSD of about 41 cm ^{-1}. Concerning the computational time, the hybrid B2PLYP//MMFF94 approach considerably outperforms the full B3LYP/SNSD: for the larger molecule of our set (a dipeptide containing 47 atoms), the anharmonic corrections are 2300 times faster using hybrid MMFF94 rather than full B3LYP, which represents an additional computation time to the harmonic calculation of merely 9 %, instead of 32100 % with the full B3LYP approach. This time-effective and accurate alternative to the traditional GVPT2 approach will allow the spectroscopy community to explore anharmonic effects in larger biomolecules, which are generally unaffordable.

### Keywords

Anharmonic corrections Hybrid QM//MM VPT2 Biomolecules Peptides Biopolymers Spectroscopy## Introduction

Vibrational frequencies are routinely calculated in the harmonic approximation with most quantum mechanics software. To take account of the anharmonicity of the electronic potential, harmonic frequencies are generally scaled down by an empirical multiplicative corrective factor, with method- and basis sets-dependant optimal values. It is even sometimes proposed to use different scaling factors for different vibrational modes to better match experimental values. Despite its empirical character, this simple approach often yields reasonable vibrational frequencies for comparison with experimental values. Alternatively, one may explicitly model the anharmonicity, although at a significant computational cost, to improve the accuracy of the predictions. Several methods are available, among which are VSCF [9], VSCF-PT2 [42], cc-VSCF [21], VCI-P [13], VT2 [29], GVPT2 [3, 4]. The generalized second-order perturbation theory (GVPT2) is largely used, as it has the advantage of combining a perturbative development to deal with weakly coupled terms and a variational treatment to handle tightly coupled ones.

Such anharmonic computations are not yet routine because they are time-consuming, and rapidly become unaffordable when the size of the system increases. In all of these methods, it is necessary to know the potential energy surface (PES). This energy surface is generally approximated by a quartic force field potential using the second, cubic, and quartic derivatives of the PES computed at the equilibrium geometry. Additionally, first, second, and third derivatives of the dipole moment must be calculated if infrared intensities are required. Critically, to achieve good agreement with experiments, high-level ab initio methods such as MP2, B3LYP [7, 8, 24], B2PLYP [17] etc., are needed, thereby limiting this approach to medium-sized molecules.

For larger systems, several promising alternatives [2, 32] to the traditional full ab initio (GVPT2, VSCF,...) calculations are proposed in the literature. It consists of computing geometry and harmonic modes at a high level of theory (CCSD(T) [14, 23] or B2PLYP [17] for example), then anharmonic corrections at a lower level, such as DFT or semi-empirical methods [10, 35].

In this work, we explore an original variation of this approach using, for the first time to our knowledge, a molecular mechanic (MM) potential for the anharmonic corrections, thus opening the way to rapid anharmonic corrections in large molecules. As an MM method, we selected the MMFF94 method, which contains explicit anharmonic terms in the potential formula. This hybrid method is described in “Methods”. We then discuss the performance and precision of the hybrid B2PLYP//MMFF94 method for the prediction of vibrational frequencies of organic and biological molecules by comparison to the experimental frequencies of a set of 15 small molecules (2 to 6 atoms) and a set of 13 medium and large molecules (9 to 45 atoms) in “Assessment of the hybrid approach by comparison with experiment”. In “Comparison to others methods”, we compare the accuracy of our hybrid method to others, known in literature, and recently applied to a few molecules studied here. In “Anharmonic accuracy and reliable determination of the harmonic frequencies”, we explore the influence of the choice of the method employed in the first phase of the hybrid approach. Finally, the timings are discussed in “Timings”.

## Methods

### The GVPT2 method

**Q**and V denote

*f*normal coordinates (

*f*=3

*N*-6 for non-linear molecule and 3

*N*-5 for linear one) and adiabatic potential energy surface (PES) terms, respectively. In our implementation, the vibration-rotation coupling terms are neglected. The adiabatic PES is approximated in terms of three coordinates coupling and developed up to the quartic term. This is a three-modes coupling representation of the potential energy surface (3MR-PES). In this approximation, the potential is a quartic force field (QFF) given by,

*V*

_{0},

*h*

_{i},

*t*

_{ijk}and

*u*

_{ijkl}denote the energy and its second-, third-, and fourth-order derivatives with respect to the normal coordinates at the equilibrium geometry, respectively. The derivatives are calculated through numerical differentiations of the energy (see Yagi et al. [43]), which allows using any ab initio or (semi-)empirical method implemented in any quantum chemistry software even if the analytical gradient is not implemented. To compute all the terms in the QFF potential, 1+6

*f*

^{2}+8

*f*(

*f*-1)(

*f*-2)/6 single point calculations are required (see Fig. 1). This makes our code highly parallelizable. In the approach of Barone et al. [3, 4], the derivatives are calculated by numerical differentiations of elements of the Hessian matrices instead, hence 6

*f*-11 points are required.

Anharmonic calculations are performed in three steps. Firstly, the harmonic frequencies and normal modes are calculated using a high-level DFT method. Secondly, the cubic and fourth derivatives are calculated using a low-level DFT or MMFF94 potentials. Finally, the anharmonic fundamental frequencies can be calculated using either VPT2 (without any treatment of resonances), GVPT2 (Fermi resonances are treated variationally), or non-variational approaches such as DCPT2 or HDCPT2 methods.

For GVPT2 calculation, the Martin et al. [25] criteria were chosen. DCPT2 is a non-variational alternative approach to GVPT2, which does not use any empirical parameter to detect the resonance terms [29]. However, this method may give poor results far from resonance. To address this failure, the HDCPT2 approach was proposed by Barone et al. [3]. It consists of mixing the DCPT2 and the standard VPT2 approaches using a transition function without the need for identifying the resonant terms.

### Technical details

Full B2PLYP GVPT2: all the values are calculated with B2PLYP/def2-TZVPP double hybrid functional (Gaussian software).

Full B3LYP GVPT2: all the values are calculated using the B3LYP/SNSD DFT hybrid functional (Gaussian software).

Hybrid MMFF94 GVPT2: our hybrid method where B2PLYP/def2-TZVPP was used to compute the harmonic modes (using Orca for large molecules and Gaussian for small molecules); and MMFF94 was used to compute the cubic and fourth derivatives of energy (using our interface to OpenBabel library). The frequencies are finally calculated via the GVPT2 method.

Hybrid MMFF94 HDCPT2: our hybrid method, using the HDCPT2 method to compute the frequencies in the last step.

Full MMFF94: all the derivatives are calculated with MMFF94 potential (via our code interfacing to OpenBabel) and the frequencies are calculated using GVPT2 method.

### Data sets

_{4}), 2 protonated monosaccharides (

*ß*GlcNMe: methyl-2-amino-2-deoxy-

*ß*-D-glucose and GlcN-6S: Glucosamine 6-sulfate), two amino acids (NATA:

*N*-acetyl tryptophan amide in two conformations referred to as C5 and C7), three dipeptides(GlyGlyH

^{+}, GlyLysH

^{+}and AVPO:Ac-Val-Phe-OMe) and 9-methyl adenine. These molecules are chosen to cover vibrational modes commonly found in biological molecules. A total of 164 experimental frequencies are included in this set, which is a total of 237 experimental frequencies. Note that our benchmark set contains only experimental vibrational frequencies. Data obtained in the liquid phase or in an argon matrix were discarded.

Fundamental anharmonic frequencies obtained with full B2PLYP, full B3LYP, hybrid MMFF94 GVPT2, and hybrid MMFF94 HDCPT2 compared to the corresponding experimental frequencies for the FS15 set of molecules. All values are given in cm ^{-1}

Molecule | Expt. | Full | Full | Hybrid | Hybrid | Molecule | Expt. | Full | Full | Hybrid | Hybrid |
---|---|---|---|---|---|---|---|---|---|---|---|

B2PLYP | B3LYP | MMFF94 | MMFF94 | B2PLYP | B3LYP | MMFF94 | MMFF94 | ||||

GVPT2 | HDCPT2 | GVPT2 | HDCPT2 | ||||||||

| 1828 | 1823 | 1846 | 1837 | 1837 |
| 625 | 623 | 621 | 623 | 623 |

CO | 2143 | 2135 | 2182 | 2142 | 2142 | 638 | 641 | 629 | 637 | 640 | |

HF | 3959 | 3954 | 3914 | 3938 | 3938 | 1033 | 1038 | 1025 | 1035 | 1034 | |

| 2331 | 2318 | 2426 | 2344 | 2344 | 1105 | 1096 | 1103 | 1102 | 1102 | |

| 775 | 767 | 793 | 773 | 773 | 1229 | 1298 | 1281 | 1299 | 1277 | |

| 1167 | 1189 | 1171 | 1185 | 1184 | 1387 | 1384 | 1368 | 1392 | 1391 | |

1249 | 1253 | 1233 | 1258 | 1257 | 1770 | 1771 | 1784 | 1771 | 1772 | ||

1500 | 1512 | 1491 | 1516 | 1516 | 2943 | 2929 | 2903 | 2945 | 2936 | ||

1746 | 1761 | 1790 | 1762 | 1762 | 3570 | 3551 | 3536 | 3605 | 3605 | ||

2782 | 2773 | 2736 | 2789 | 2788 |
| 826 | 837 | 824 | 843 | 836 | |

2843 | 2844 | 2813 | 2854 | 2816 | 949 | 969 | 955 | 931 | 931 | ||

| 1595 | 1592 | 1580 | 1603 | 1602 | 1023 | 1048 | 1031 | 989 | 989 | |

3657 | 3661 | 3645 | 3700 | 3698 | 1342 | 1360 | 1350 | 1367 | 1367 | ||

3756 | 3758 | 3741 | 3784 | 3784 | 1444 | 1454 | 1433 | 1460 | 1459 | ||

| 950 | 970 | 965 | 889 | 889 | 1623 | 1645 | 1653 | 1647 | 1668 | |

1627 | 1633 | 1616 | 1642 | 1637 | 2989 | 3004 | 2976 | 2990 | 2970 | ||

1627 | 1633 | 1616 | 1642 | 1637 | 3026 | 3038 | 3006 | 3030 | 3030 | ||

3337 | 3361 | 3329 | 3409 | 3397 | 3103 | 3096 | 3058 | 3086 | 3085 | ||

3444 | 3459 | 3428 | 3476 | 3476 | 3106 | 3120 | 3084 | 3109 | 3109 | ||

3444 | 3459 | 3429 | 3476 | 3475 |
| 523 | 489 | 500 | 534 | 586 | |

| 518 | 509 | 485 | 512 | 512 | 1599 | 1600 | 1581 | 1592 | 1579 | |

1151 | 1121 | 1090 | 1122 | 1122 | 1616 | 1611 | 1588 | 1603 | 1598 | ||

1362 | 1326 | 1242 | 1322 | 1322 | 3601 | 3582 | 3537 | 3627 | 3624 | ||

| 878 | 903 | 914 | 922 | 922 | 3660 | 3648 | 3622 | 3658 | 3656 | |

1274 | 1271 | 1277 | 1277 | 1277 | 3735 | 3728 | 3693 | 3720 | 3719 | ||

1394 | 1397 | 1400 | 1409 | 1409 | 3745 | 3740 | 3718 | 3728 | 3726 | ||

3615 | 3610 | 3586 | 3592 | 3591 |
| 1033 | 1028 | 1022 | 1034 | 1034 | |

3614 | 3606 | 3585 | 3645 | 3644 | 1060 | 1074 | 1057 | 1074 | 1074 | ||

| 1058 | 1064 | 1055 | 1073 | 1073 | 1345 | 1333 | 1310 | 1350 | 1350 | |

1061 | 1085 | 1070 | 1050 | 1050 | 1455 | 1461 | 1444 | 1466 | 1455 | ||

1127 | 1142 | 1124 | 1107 | 1107 | 1477 | 1472 | 1453 | 1470 | 1446 | ||

1344 | 1344 | 1332 | 1367 | 1366 | 1477 | 1482 | 1461 | 1486 | 1504 | ||

1452 | 1468 | 1455 | 1470 | 1469 | 2844 | 2842 | 2809 | 2835 | 2907 | ||

1638 | 1659 | 1675 | 1663 | 1663 | 2960 | 2922 | 2886 | 2901 | 2960 | ||

2914 | 2887 | 2848 | 2889 | 2902 | 3000 | 3003 | 2968 | 3033 | 3025 | ||

3025 | 2963 | 2944 | 2980 | 3018 | 3681 | 3679 | 3649 | 3704 | 3704 | ||

3263 | 3302 | 3278 | 3311 | 3310 |

Fundamental anharmonic frequencies obtained with full B3LYP, hybrid MMFF94 GVPT2, and hybrid MMFF94 HDCPT2 compared to the corresponding experimental frequencies for the FL13 set of molecules. All values are given in cm ^{-1}

Molecule | Mode | Expt. | Full | Hybrid | Hybrid | Molecule | Mode | Expt. | Full | Hybrid | Hybrid |
---|---|---|---|---|---|---|---|---|---|---|---|

B3LYP | MMFF94 | MMFF94 | B3LYP | MMFF94 | MMFF94 | ||||||

GVPT2 | HDCPT2 | GVPT2 | HDCPT2 | ||||||||

NATA-C5 | NH | 3538 | 3529 | 3580 | 3580 | Benzene | CH str s | 3074 | 3049 | 3054 | 3055 |

Indole NH str | 3523 | 3507 | 3533 | 3533 | ring s d | 993 | 1009 | 1010 | 1010 | ||

f-Amide NH str | 3430 | 3413 | 3470 | 3470 | CH b s | 1350 | 1349 | 1382 | 1364 | ||

NH | 3417 | 3402 | 3475 | 3471 | ring d oop | 674 | 677 | 696 | 696 | ||

Phe CH str | 3074 | 3051 | 3057 | 3067 | CH str as | 3057 | 2997 | 3011 | 3057 | ||

Phe CH str | 3059 | 3044 | 3049 | 3059 | ring d | 1010 | 997 | 1026 | 1026 | ||

Alkyl CH str | 3004 | 2970 | 2997 | 2996 | CH oopb | 990 | 985 | 924 | 924 | ||

Alkyl CH str | 2956 | 2936 | 2972 | 2971 | CH s oopb | 707 | 684 | 671 | 671 | ||

Alkyl CH str | 2933 | 2928 | 2963 | 2944 | CC str | 1309 | 1325 | 1347 | 1347 | ||

CH b as | 1150 | 1157 | 1175 | 1175 | |||||||

NATA-C7 | NH | 3516 | 3492 | 3542 | 3541 | CH oopb | 847 | 844 | 829 | 829 | |

Indole NH str | 3521 | 3498 | 3531 | 3531 | CH oopb | 847 | 847 | 831 | 831 | ||

f-Amide NH str | 3429 | 3422 | 3475 | 3474 | CH str as | 3047 | 3028 | 3044 | 3068 | ||

NH | 3334 | 3302 | 3385 | 3383 | CH str as | 3047 | 3027 | 3044 | 3068 | ||

Phe CH str | 3092 | 3052 | 3066 | 3060 | CH b as | 1484 | 1478 | 1509 | 1509 | ||

Phe CH str | 3072 | 3043 | 3059 | 3058 | CH b as | 1484 | 1480 | 1510 | 1510 | ||

Phe CH str | 3048 | 3040 | 3038 | 3038 | CH b s | 1038 | 1037 | 1050 | 1050 | ||

Alkyl CH str | 2999 | 2954 | 2992 | 2992 | CH b s | 1038 | 1039 | 1051 | 1051 | ||

Alkyl CH str | 2979 | 2975 | 2930 | 2927 | CH str as | 3057 | 3033 | 3039 | 3040 | ||

Alkyl CH str | 2938 | 2933 | 2972 | 2956 | CH str as | 3057 | 3033 | 3040 | 3039 | ||

Alkyl CH str | 2919 | 2899 | 2936 | 2935 | CC str | 1601 | 1592 | 1622 | 1628 | ||

CC str | 1601 | 1592 | 1623 | 1630 | |||||||

AVPO | Phe NH str | 3451 | 3432 | 3473 | 3473 | CH b as | 1178 | 1177 | 1198 | 1198 | |

Val NH str | 3441 | 3408 | 3476 | 3475 | CH b as | 1178 | 1179 | 1200 | 1200 | ||

Phe CH str | 3096 | 3053 | 3064 | 3060 | ring d | 608 | 612 | 618 | 618 | ||

Phe CH str | 3076 | 3045 | 3046 | 3045 | ring d | 608 | 612 | 619 | 619 | ||

Phe CH str | 3038 | 3029 | 3052 | 3063 | CH oopb | 976 | 972 | 919 | 919 | ||

Phe CH str | 3006 | 3020 | 3022 | 3009 | CH oopb | 976 | 973 | 921 | 921 | ||

Met(Val) CH str | 2974 | 2965 | 2997 | 2997 | ring d oop | 398 | 402 | 406 | 406 | ||

Met(Val) CH str | 2965 | 2953 | 2981 | 2981 | ring d oop | 398 | 403 | 407 | 407 | ||

Met(Val) CH str | 2941 | 2879 | 2985 | 2985 | |||||||

Phe CO str | 1765 | 1752 | 1746 | 1746 | DMSO | O-S-O str s | 758 | 663 | 724 | 725 | |

Ace CO str | 1711 | 1704 | 1700 | 1702 | O-S-O str as | 814 | 721 | 787 | 791 | ||

Val CO str | 1696 | 1688 | 1686 | 1684 | CO2 str as | 1006 | 969 | 1000 | 1000 | ||

O=S=O str s | 1206 | 1093 | 1200 | 1200 | |||||||

9M-Adenine | NH | 1632 | 1629 | 1645 | 1644 | O=S=O str as | 1410 | 1293 | 1417 | 1417 | |

NH | 1599 | 1584 | 1596 | 1598 | |||||||

NH | 1515 | 1498 | 1513 | 1516 | Cyclopropane | CH | 3027 | 3011 | 3058 | 3045 | |

NH | 1470 | 1463 | 1493 | 1493 | CH | 1499 | 1439 | 1532 | 1529 | ||

d CH | 1450 | 1434 | 1527 | 1532 | ring str s | 1189 | 1186 | 1202 | 1202 | ||

CH | 1429 | 1428 | 1433 | 1421 | CH | 1127 | 1113 | 1156 | 1156 | ||

CH | 1414 | 1401 | 1412 | 1412 | CH | 1067 | 1056 | 1091 | 1091 | ||

CH b, NH | CH | 3102 | 3070 | 3104 | 3104 | ||||||

CH b, CN str | 1369 | 1369 | 1386 | 1386 | CH | 854 | 852 | 848 | 848 | ||

rings d, CN str, d CH | 1345 | 1331 | 1353 | 1353 | CH | 3019 | 3003 | 3044 | 3032 | ||

CH b, CN str | 1327 | 1334 | 1350 | 1350 | CH | 3019 | 3004 | 3045 | 3033 | ||

ring d, NH | 1292 | 1309 | 1321 | 1323 | CH | 1440 | 1428 | 1452 | 1452 | ||

NH | 1256 | 1246 | 1254 | 1251 | CH | 1440 | 1430 | 1453 | 1453 | ||

CH b, NH b | 1232 | 1235 | 1246 | 1246 | CH | 1028 | 1018 | 1046 | 1046 | ||

CH b, NH | 1199 | 1190 | 1203 | 1203 | CH | 1028 | 1024 | 1046 | 1046 | ||

d CH | 1136 | 1121 | 1179 | 1179 | CC str s | 868 | 855 | 876 | 877 | ||

NH | 1067 | 1049 | 1074 | 1074 | CC str s | 868 | 855 | 878 | 878 | ||

NH | 1036 | 1034 | 1057 | 1057 | CH | 3082 | 3047 | 3085 | 3085 | ||

NH | 1000 | 970 | 988 | 988 | CH | 3082 | 3048 | 3086 | 3086 | ||

CH oopb | 958 | 949 | 958 | 957 | CH | 1191 | 1182 | 1198 | 1198 | ||

rings d, CH b, CH | 895 | 894 | 904 | 904 | CH | 1191 | 1181 | 1199 | 1199 | ||

CH oopb | 841 | 821 | 826 | 826 | CH | 738 | 732 | 731 | 725 | ||

rings tors, CH oopb | 800 | 792 | 803 | 803 | CH | 738 | 738 | 736 | 727 | ||

rings d, CN str | 730 | 736 | 736 | 737 | |||||||

rings d, d CH | 715 | 718 | 727 | 727 | Pyruvic acid | OH str | 3463 | 3411 | 3511 | 3509 | |

rings tors | 673 | 675 | 689 | 689 | CH | 3025 | 3003 | 3036 | 3035 | ||

rings tors, CH oopb | 640 | 644 | 656 | 656 | CH | 2941 | 2925 | 2949 | 2925 | ||

NH | 577 | 573 | 582 | 584 | C3=O str | 1804 | 1811 | 1800 | 1800 | ||

rings tors, CH oopb | 553 | 555 | 567 | 567 | C2=O str | 1737 | 1750 | 1725 | 1730 | ||

CH | 1424 | 1417 | 1442 | 1441 | |||||||

GlcN-6S | SO2 str s | 1180 | 1082 | 1198 | 1200 | CC str as | 1391 | 1373 | 1387 | 1388 | |

SO(H) str | 885 | 775 | 859 | 861 | CH | 1360 | 1322 | 1351 | 1343 | ||

SO(C) str | 780 | 986 | 771 | 772 | COH b | 1211 | 1197 | 1221 | 1221 | ||

NH\(_{3}^{+}\) str as | 3256 | 3245 | 3329 | 3307 | CO str | 1133 | 1126 | 1144 | 1141 | ||

NH\(_{3}^{+}\) str as | 3333 | 3321 | 3378 | 3377 | CH | 970 | 960 | 976 | 976 | ||

OH(3) str | 3600 | 3468 | 3656 | 3655 | CC str s | 761 | 747 | 757 | 758 | ||

(S)OH str | 3582 | 3537 | 3633 | 3632 | C2=O b | 604 | 601 | 604 | 604 | ||

OH(4) str | 3550 | 3575 | 3588 | 3588 | CH | 1030 | 1009 | 1030 | 1030 | ||

OH tors | 668 | 675 | 671 | 669 | |||||||

| NH\(_{3}^{+}\) str s | 3242 | 3241 | 3314 | 3286 | ||||||

NH\(_{3}^{+}\) str as | 3297 | 3287 | 3338 | 3338 | Methyloxyrane | CH | 3051 | 3015 | 3053 | 3053 | |

NH\(_{3}^{+}\) str as | 3317 | 3308 | 3357 | 3356 | CH str | 3001 | 2968 | 3001 | 3001 | ||

OH(4) str | 3555 | 3496 | 3602 | 3602 | CH | 2995 | 2957 | 2989 | 2988 | ||

OH(3) str | 3612 | 3588 | 3659 | 3658 | CH | 2974 | 2943 | 2979 | 2979 | ||

OH(6) str | 3672 | 3672 | 3717 | 3717 | CH | 2942 | 2929 | 2862 | 2959 | ||

GlyLysH | OH str | 3584 | 3535 | 3609 | 3608 | GlyGlyH | OH str | 3584 | 3526 | 3605 | 3605 |

NH str as | 3470 | 3359 | 3419 | 3419 | OH str | 3584 | 3542 | 3610 | 3610 | ||

NH\(_{3}^{+}\) str | 3426 | 3350 | 3410 | 3409 | NH\(_{3}^{+}\) str s | 3372 | 3170 | 3346 | 3344 | ||

NH\(_{3}^{+}\) str | 3410 | 3358 | 3400 | 3400 | NH str s | 3400 | 3333 | 3389 | 3379 | ||

NH str s | 3371 | 3310 | 3377 | 3367 | CH str s | 3045 | 2961 | 3007 | 2994 | ||

NH\(_{3}^{+}\) str | 3300 | 3188 | 3278 | 3283 | CH str s | 3000 | 2942 | 2985 | 2965 | ||

NH\(_{3}^{+}\) str | 3138 | 2976 | 3092 | 3097 | CH str as | 3042 | 3002 | 2981 | 2981 |

## Results and discussion

In order to validate the performance of our hybrid MMFF94 GVPT2 approach, we performed the calculation of the fundamental frequencies on both sets FS15 and FL13. To compare the accuracy of our method to that of the standard GVPT2 approaches often used in the literature (full B2PLYP GVPT2 and full B3LYP GVPT2), we also calculated the frequencies with these methods. The full B2PLYP GVPT2 method is known as a very accurate method to study small- and medium-sized molecules but the computational cost rapidly becomes prohibitive for larger molecules. The full B3LYP GVPT2, with the SNSD basis set, was proposed by Barone et al. as a good compromise between accuracy and efficiency for larger molecules.

### Assessment of the hybrid approach by comparison with experiment

^{-1}, a AUE of 19 cm

^{-1}, and a maximum error of 72 cm

^{-1}, which corresponds to the NH

_{3}symmetric mode. The second largest error is the OH bending mode in CH

_{2}

*O*

_{2}(70 cm

^{-1}). Similar results are obtained using the hybrid MMFF94 HDCPT2 approach. For full B2PLYP calculation, the RMSD, AUE, and MaxUE are 19 cm

^{-1}, 13 cm

^{-1}, 69 cm

^{-1}, respectively, again corresponding to the OH bending mode in CH

_{2}

*O*

_{2}. The full B3LYP calculation reproduces the fundamental frequencies with a RMSD of 34 cm

^{-1}and a AUE of 26 cm

^{-1}. Here, the maximum error

Statistics (MaxUE, AUE, and RMSD) for each molecule of both sets, then for each set and finally for all molecules. All values are given in cm ^{-1}

Molecule | Stat | Full | Full | Hybrid | Hybrid | Molecule | Stat | Full | Hybrid | Hybrid |
---|---|---|---|---|---|---|---|---|---|---|

B2PLYP | B3LYP | MMFF94 | MMFF94 | B3LYP | MMFF94 | MMFF94 | ||||

GVPT2 | HDCPT2 | GVPT2 | HDCPT2 | |||||||

| MaxUE | 5 | 18 | 9 | 9 | NATA-C5 | MaxUE | 34 | 58 | 54 |

AUE | 5 | 18 | 9 | 9 | AUE | 17 | 26 | 21 | ||

RMSD | 5 | 18 | 9 | 9 | RMSD | 19 | 31 | 28 | ||

CO | MaxUE | 8 | 39 | 1 | 1 | NATA-C7 | MaxUE | 45 | 51 | 52 |

AUE | 8 | 39 | 1 | 1 | AUE | 22 | 26 | 25 | ||

RMSD | 8 | 39 | 1 | 1 | RMSD | 26 | 31 | 30 | ||

HF | MaxUE | 5 | 45 | 21 | 21 | AVPO | MaxUE | 62 | 44 | 44 |

AUE | 5 | 45 | 21 | 21 | AUE | 22 | 23 | 23 | ||

RMSD | 5 | 45 | 21 | 21 | RMSD | 27 | 25 | 25 | ||

| MaxUE | 13 | 95 | 13 | 13 | 9M-adenine | MaxUE | 30 | 77 | 82 |

AUE | 13 | 95 | 13 | 13 | AUE | 9 | 14 | 15 | ||

RMSD | 13 | 95 | 13 | 13 | RMSD | 12 | 21 | 22 | ||

| MaxUE | 8 | 18 | 2 | 2 | GlcN-6S | MaxUE | 206 | 73 | 55 |

AUE | 8 | 18 | 2 | 2 | AUE | 80 | 39 | 36 | ||

RMSD | 8 | 18 | 2 | 2 | RMSD | 103 | 44 | 40 | ||

| MaxUE | 22 | 46 | 18 | 27 |
| MaxUE | 59 | 72 | 47 |

AUE | 10 | 25 | 13 | 15 | AUE | 17 | 49 | 44 | ||

RMSD | 13 | 30 | 13 | 16 | RMSD | 26 | 50 | 44 | ||

| MaxUE | 4 | 15 | 43 | 41 | GlyLysH | MaxUE | 162 | 51 | 51 |

AUE | 3 | 14 | 26 | 26 | AUE | 89 | 25 | 23 | ||

RMSD | 3 | 14 | 30 | 29 | RMSD | 97 | 30 | 28 | ||

| MaxUE | 24 | 16 | 72 | 61 | Benzene | MaxUE | 60 | 66 | 66 |

AUE | 14 | 13 | 38 | 34 | AUE | 11 | 23 | 23 | ||

RMSD | 16 | 13 | 44 | 40 | RMSD | 16 | 28 | 27 | ||

| MaxUE | 36 | 120 | 40 | 40 |
| MaxUE | 117 | 34 | 33 |

AUE | 25 | 71 | 25 | 25 | AUE | 91 | 16 | 15 | ||

RMSD | 27 | 80 | 29 | 29 | RMSD | 96 | 20 | 19 | ||

| MaxUE | 25 | 36 | 44 | 44 | Cyclopropane | MaxUE | 60 | 33 | 30 |

AUE | 9 | 21 | 23 | 23 | AUE | 16 | 14 | 13 | ||

RMSD | 12 | 25 | 27 | 27 | RMSD | 21 | 17 | 15 | ||

| MaxUE | 62 | 81 | 48 | 47 | Pyruvic-acid | MaxUE | 52 | 48 | 46 |

AUE | 23 | 26 | 25 | 20 | AUE | 17 | 10 | 10 | ||

RMSD | 29 | 38 | 28 | 23 | RMSD | 21 | 15 | 15 | ||

| MaxUE | 69 | 52 | 70 | 48 | Methyloxyrane | MaxUE | 38 | 80 | 17 |

AUE | 14 | 20 | 14 | 11 | AUE | 30 | 19 | 6 | ||

RMSD | 25 | 26 | 26 | 20 | RMSD | 31 | 36 | 9 | ||

| MaxUE | 25 | 45 | 34 | 45 | GlyGlyH | MaxUE | 202 | 61 | 61 |

AUE | 15 | 17 | 16 | 19 | AUE | 79 | 28 | 35 | ||

RMSD | 16 | 21 | 19 | 23 | RMSD | 95 | 32 | 38 | ||

| MaxUE | 34 | 64 | 26 | 63 | |||||

AUE | 12 | 34 | 13 | 23 | ||||||

RMSD | 16 | 37 | 15 | 29 | ||||||

| MaxUE | 38 | 74 | 59 | 63 | |||||

AUE | 9 | 27 | 17 | 19 | ||||||

RMSD | 14 | 33 | 24 | 26 | ||||||

FS15 | MaxUE | 69 | 120 | 72 | 63 | FL13 | MaxUE | 206 | 80 | 82 |

AUE | 13 | 26 | 19 | 20 | AUE | 26 | 21 | 20 | ||

RMSD | 19 | 34 | 25 | 25 | RMSD | 44 | 28 | 26 | ||

FS15 + FL13 | MaxUE | 206 | 80 | 82 | ||||||

AUE | 26 | 21 | 20 | |||||||

RMSD | 41 | 27 | 26 |

(120 cm ^{-1})is observed for the stretching asymmetric mode in SO _{2} molecule. The RMSD of the hybrid MMFF94 method is only 6 cm ^{-1} larger than this of full B2PLYP, while the RMSD of full B3LYP is nearly twice this of full B2PLYP.

^{-1}for individual molecules of the set. The RMSD calculated on the whole set is 28 cm

^{-1}, which is consistent with the precision obtained for small molecules. The larger errors are obtained for the CH

_{3}stretching symmetric mode in methyloxyrane (MaxUE=80 cm

^{-1}), the NH\(_{3}^{+}\) (72 cm

^{-1}) and the OH stretching modes (36 cm

^{-1}) in

*ß*GlcNMe molecule. For the full B3LYP calculation, the individual RMSD ranges from 12 cm

^{-1}for 9M-Adenine to 103 cm

^{-1}for GlcN-6S while the RMSD of the set is 44 cm

^{-1}with a maximal deviation of 206 cm

^{-1}for the SO(C) stretching mode of GlcN-6S.

Overall, 76 % of calculated hybrid MMFF94 frequencies have a deviation between -30 and +30 cm ^{-1} to be compared to 71 % for the full B3LYP calculations.

^{-1}. The second range, above 2000 cm

^{-1}concerns the X-H stretching modes involving a hydrogen atom. Figure 6 shows the statistical parameters for the two ranges of frequencies. It is clear that the wavenumbers of the first range are better reproduced than these of the second range, whatever the approach: our hybrid MMFF94 or that of full B3LYP. The RMSD with our approach are 22, 33, and 27 cm

^{-1}for low frequencies, high frequencies, and all frequencies, respectively. They are 34, 49, and 41 cm

^{-1}for the full B3LYP. Thus, for the two ranges of wavenumbers our hybrid approach outperforms the full B3LYP.

### Comparison to others methods

Several molecules of our sets have already been studied in the literature using hybrid VSCF-PT2 or hybrid GVPT2 approaches. NATA-C5, NATA-C7, and AVPO were studied recently by Roy et al. [36] using hybrid MP2//HF potentials and the VSCF-PT2 method to compute the fundamental frequencies. In this approach, the harmonic frequencies are calculated at MP2/cc-pvdz level of theory and the pair-wise coupling terms are calculated at HF/cc-pvdz level (579960 single-point HF calculations are needed for AVPO). This hybrid approach requires certainly more computing time than our hybrid approach, where the cubic and quartic derivatives are calculated using a molecular mechanics potential. Compared to experimental values, the frequencies obtained with MP2//HF VSCF-PT2 are reproduced with a RMSD of 22 cm ^{-1}, 33 cm ^{-1}, and 38 cm ^{-1} for NATA-C5, NATA-C7, and AVPO, respectively, to be compared to 31 cm ^{-1}, 31 cm ^{-1}, and 25 cm ^{-1} using the hybrid MMFF94 GVPT2 approach. It is clear, statistically speaking, that our hybrid approach and the MP2//HF VSCF-PT2 approach have comparable accuracy.

The anharmonic frequencies of the two proton-bound amino acid dimers GlyGlyH ^{+} and GlyLysH ^{+} have been reported by Adesokan et al. [1] using a hybrid MP2//PM3 method. In this approach, the potential surface is calculated using PM3 semi-empirical method but the normal coordinates are scaled to reproduce the MP2 harmonic frequencies. Using this potential, the frequencies are then obtained using the VSCF-PT2 method. This approach reproduces the experimental values with a RMSD of 66 cm ^{-1} for GlyGlyH ^{+} and 68 cm ^{-1} for GlyLysH ^{+} to be compared to the RMSD of 32 cm ^{-1} and 30 cm ^{-1} obtained with our approach. A maximum deviation of 168 cm ^{-1} is obtained for the NH\(_{3}^{+}\) stretching mode for GlyLysH ^{+} molecule, with MP2//PM3, while a maximum deviation of 61 cm ^{-1} is observed for a CH asymmetric stretching mode of GlyGlyH ^{+} molecule in our approach. The hybrid MMFF94 approach is thus faster and more accurate than hybrid MP2//PM3 VSCF-PT2. Cyclopropane [32], pyruvic acid [6], and methyloxyrane [5] were recently studied by Barone and coworkers using GVPT2 and B2PLYP/aug-cc-pvtz potential (for harmonic and anharmonic parts). Using this method, the experimental frequencies were reproduced with a RMSD of 8, 12, and 12 cm ^{-1}, respectively, to be compared to 17, 15, and 36 cm ^{-1} using the hybrid MMFF94 GVPT2. A hybrid CCSD(T)//B2PLYP GVPT2 was proposed by these authors to further reduce the errors, however, it is important to note that this alternative requires more computing resources than the full B2PLYP method and is thus not suitable to study large molecules(> 30 atoms).

### Anharmonic accuracy and reliable determination of the harmonic frequencies

We found above that using the molecular mechanic MMFF94 method to calculate the cubic and quartic derivatives of the potential, following the ab initio simulation of the equilibrium geometry and second derivatives of the potential, yielded very good results as compared to previously reported methods. Therefore, we further considered an alternative *full molecular mechanic approach*, where all properties are calculated using the MMFF94 potential. In other words, is it necessary to compute the equilibrium geometry and harmonic modes using a high precision ab initio method? To answer this question, we calculated the equilibrium geometry and all derivatives using MMFF94 potential. The anharmonic frequencies are then calculated and compared to experimental ones. The RMSD obtained is 180 cm ^{-1} for full MMFF94 to be compared to 25 cm ^{-1} obtained with MMFF94 hybrid for the set FS15. It is clear that the full MMFF94 approach gives poor results as compared to hybrid MMFF94. Specifically, the harmonic frequencies calculated with the MMFF94 potential are very inaccurate, as compared to the B2PLYP value. The anharmonic terms derived from the MMFF94 potential, however, are accurate, which explains the good result obtained with the hybrid method. In fact, this remark is not specific to our hybrid approach. To obtain very good anharmonic frequencies, it is important to compute the harmonic ones using a high-level method of theory with a very large basis set while the anharmonic parts can be calculated with a low-level method or a smaller basis set [12]. This could explain why our hybrid B2PLYP//MMFF94 is more accurate than the full B3LYP method. Indeed, in our hybrid approach, the B2PLYP/def2-TZVPP method is used to compute the harmonic frequencies, which is certainly more accurate than the B3LYP/SNSD method.

### Timings

Total CPU time (in h) for full B3LYP, hybrid MMFF94 GVPT2 and hybrid MMFF94 HDCPT2 and the percentage of each step of the anharmonic calculation in the total CPU time for the five biggest molecules of FL13

Molecule | GlcN-6S | NATA-C5 | NATA-C7 | GlyLysH | AVPO |
---|---|---|---|---|---|

Number of atoms | 30 | 33 | 33 | 35 | 47 |

Total CPU time full B3LYP | 1967 | 3575 | 3925 | 3801 | 23177 |

% second derivatives | 0.47 | 1 | 0.62 | 0.28 | 0.31 |

% third and fourth derivatives | 99.52 | 99 | 99.36 | 99.69 | 99.65 |

% GVPT2 | 0.01 | 0 | 0.02 | 0.03 | 0.04 |

Total CPU time hybrid MMFF94 GVPT2 | 22 | 34 | 34 | 30 | 133 |

% second derivatives | 98.35 | 96.42 | 98.05 | 94.61 | 92.31 |

% third and fourth derivatives | 0.05 | 0.06 | 0.05 | 0.07 | 0.07 |

% GVPT2 | 1.59 | 3.53 | 1.89 | 5.32 | 7.61 |

Total CPU time hybrid MMFF94 HDCPT2 | 22 | 32 | 34 | 28 | 123 |

% second derivatives | 99.87 | 99.86 | 99.87 | 99.80 | 99.83 |

% third and fourth derivatives | 0.05 | 0.06 | 0.06 | 0.07 | 0.08 |

% HDCPT2 | 0.08 | 0.08 | 0.08 | 0.12 | 0.09 |

From this table, it is clear that the hybrid MMFF94 is much faster than a full B3LYP GVPT2 calculation and the efficiency increases with the size of system. For example, the calculation of third and fourth derivatives with the hybrid MMFF94 method is 248342 times faster than full B3LYP GVPT2 for AVPO. Indeed, the post-harmonic calculation for AVPO requires 962 days with full B3LYP GVPT2, and only 10 h with hybrid MMFF94 for a comparable accuracy.

## Conclusions

To take explicit account of the anharmonicity, it is essential to go beyond the use of empirical scaling factors. GVPT2 has become a popular approach, but its computational cost increases exponentially with the size of the system, hence limiting its applicability to small- and medium-sized molecules. In order to break through this size limitation, we explored the accuracy of a new hybrid scheme for two datasets of molecules. This study revealed that the force field MMFF94 yields quite accurate results with a RMSD of 25 cm ^{-1} for the set of small molecules, 28 cm ^{-1} for the set of large molecules and 27 cm ^{-1} for the 237 frequencies of all molecules considered in our study. This very fast hybrid method gives similar results to these obtained with a full B2PLYP/def2-TZVPP, which yields a RMSD of 19 cm ^{-1}. The MMFF94 hybrid method outperforms in accuracy and in CPU time the full B3LYP GVPT2, which yields a RMSD of 41 cm ^{-1} for the all molecules for both sets studied here.

Besides its remarkable accuracy, the method offers a considerable gain on computational time. Indeed, the anharmonic corrections with MMFF94 hybrid is orders of magnitude smaller than that of anharmonic corrections at high level. For the larger molecules, the anharmonic corrections with MMFF94 hybrid method are about 2300 times faster than with full B3LYP GVPT2. Remarkably, the anharmonic corrections are 12 times faster than the harmonic term. In other words, with the hybrid MMFF94 GVPT2 approach, about 92 % of the computational time is used for the normal modes, while only 8 % is needed for the anharmonic corrections. The hybrid MMFF94 HDCPT2 calculation can further reduce the cost of anharmonic corrections from 8 % to 1 % while maintaining very good accuracy. This time-effective and accurate alternative to traditional GVPT2 anharmonic corrections will allow the community of molecular spectroscopy to study larger biopolymers, which are generally unaffordable at the standard GVPT2 level of theory.

## Notes

### Acknowledgments

This work was granted access to the HPC resour ces of the FLMSN, “Fédération Lyonnaise de Modélisation et Sciences Numériques”, partner of EQUIPEX EQUIP@MESO, and to the “Centre de calcul CC-IN2P3” at Villeurbanne, France. The authors are members of the Glycophysics Network (http://www.glyms.univ-lyon1.fr).

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