Journal of Molecular Modeling

, 22:285 | Cite as

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

  • Loïc Barnes
  • Baptiste Schindler
  • Isabelle Compagnon
  • Abdul-Rahman Allouche
Original Paper
Part of the following topical collections:
  1. Festschrift in Honor of Henry Chermette

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

All anharmonic vibrational frequencies have been calculated using the generalized second-order vibrational perturbation theory (GVPT2) model or its variants DCPT2 and HDCPT2 [3, 4, 29]. To compute anharmonic IR spectra, we implemented a code (written in C) with an interface to ORCA [28], Gaussian [16], OpenMopac [38] and other computational chemistry packages. We describe here only the important points of the theory in order to highlight the differences between our implementation and the widely used version of V. Barone [3, 4] in the Gaussian suite of programs [16]. The vibrational Hamiltonian of a non-linear N-atom molecule can be written in terms of normal coordinates as:
$$H= -\frac{1}{2} \sum\limits_{i=1}^{i=f} {\frac{\partial^{2}}{\partial {Q_{i}^{2}}}} + V(\bold{Q}) $$
where Q and V denote f normal coordinates (f=3N-6 for non-linear molecule and 3N-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,
$$\begin{array}{@{}rcl@{}} V^{QFF}(\bold{Q}) = V_{0} + V_{1}(\bold{Q}) + V_{2}(\bold{Q}) + V_{3}(\bold{Q}) &\\ V_{1}(\bold{Q}) = \sum\limits_{i=1}^{f} { \frac{1}{2} h_{i} {Q_{i}^{2}} + \frac{1}{6} t_{iii} {Q_{i}^{3}} + \frac{1}{24} u_{iiii} {Q_{i}^{4}}}&\\ V_{2}(\bold{Q}) = \sum\limits_{ij, i \neq j}^{f} {\frac{1}{2} t_{ijj} Q_{i} {Q_{j}^{2}} + \frac{1}{6} u_{ijjj} Q_{i} {Q_{j}^{3}}} + \sum\limits_{ij,i<j}^{f} {\frac{1}{4} u_{iijj} {Q_{i}^{2}} {Q_{j}^{2}}} &\\ V_{3}(\bold{Q}) = \sum\limits_{ijk, i \neq j<k}^{f} { t_{ijk} Q_{i} Q_{j} Q_{k}} + \sum\limits_{ijk,i \neq j<k}^{f} {\frac{1}{2} u_{iijk} {Q_{i}^{2}} Q_{j} Q_{k}} \end{array} $$
where V0, hi, tijk and uijkl 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+6f2+8f(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 6f-11 points are required.
Fig. 1

Hybrid approach-pseudocode

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

The Gaussian 09 package was employed to compute B3LYP/SNSD [4, 11, 12] energies for all test molecules and to compute the B2PLYP/def2-TZVPP [41] energies for small molecules. ORCA [28] was used to calculate B2PLYP/def2-TZVPP [41] normal modes for larger molecules (only because, thanks to RIJCOSX approximation, it proves faster than Gaussian in this case). To optimize the structures and to compute the harmonic frequencies with ORCA, we used the RIJCOSX, Grid6, TightSCF keywords. With Gaussian 09, we added SCF=Tight and Opt=Tight keywords. The OpenBabel [30] library was used to compute MMFF94 [18] energies. We developed a new program with an interface to ORCA, and other computational chemistry software, to implement the GVPT2 approach [3, 4]. The third and fourth derivatives were calculated through numerical differentiations of the energy, using dimensionless reduced coordinates with a step size of 0.5 [43]. The harmonic frequencies and the corresponding modes were calculated at B2PLYP/def2-TZVPP level of theory. For third and fourth derivatives, the energies are calculated using MMFF94 potential. Finally, using these derivatives, the anharmonic frequencies are calculated via our code of GVPT2 method. For simplicity, in the next sections, we use the following notations:
  • 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.

Finally, to reduce the real time of calculation for large molecules (more than 30 atoms), the derivatives for the full B3LYP GVPT2 were calculated mode by mode, in parallel, using SelectAnharmonicModes keyword of Gaussian. These derivatives were then collected and the frequencies were finally calculated using GVPT2 method.

Data sets

To assess the performance of the hybrid MMFF94 methods, we used two sets of organic and biological molecules. The first set (FS15) given in Table 1 is a compilation of 15 small molecules (2 to 6 atoms) for which the 73 experimental frequencies are available in the literature [19, 34]. This set is designed to cover a broad range of vibrational modes for small molecules. The second dataset (FL13) given in Table 2 is a compilation of 13 medium- and large-sized molecules (9 to 45 atoms) including benzene, cyclopropane, methyloxyrane, pyruvic acid, dimethyl sulfate (DMSO 4), 2 protonated monosaccharides (ßGlcNMe: 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.
Table 1

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

C2a

1828

1823

1846

1837

1837

CH2O2b

625

623

621

623

623

COa

2143

2135

2182

2142

2142

 

638

641

629

637

640

HFa

3959

3954

3914

3938

3938

 

1033

1038

1025

1035

1034

N2a

2331

2318

2426

2344

2344

 

1105

1096

1103

1102

1102

P2a

775

767

793

773

773

 

1229

1298

1281

1299

1277

H2COb

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

C2H4b

826

837

824

843

836

 

2843

2844

2813

2854

2816

 

949

969

955

931

931

H2Ob

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

NH3b

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

H2O- H2Od

523

489

500

534

586

SO2b

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

H2O2c

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

CH3OHb

1033

1028

1022

1034

1034

 

3614

3606

3585

3645

3644

 

1060

1074

1057

1074

1074

CH2NHb

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

      

Experimental values taken from a) Ref. [19], b) Ref. [20], c) Ref. [34], d) Ref. [22]

Table 2

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

Modea

Expt.

Full

Hybrid

Hybrid

Molecule

Mode

Expt.

Full

Hybrid

Hybrid

 

B3LYP

MMFF94

MMFF94

 

B3LYP

MMFF94

MMFF94

    

GVPT2

HDCPT2

    

GVPT2

HDCPT2

NATA-C5b

NH 2 str as

3538

3529

3580

3580

Benzeneg

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 2 str s

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-C7b

NH 2 str as

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 2 str s

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

AVPOc

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 4h

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-Adenined

NH 2 b, CC str

1632

1629

1645

1644

 

O=S=O str as

1410

1293

1417

1417

 

NH 2 b, CC str

1599

1584

1596

1598

      
 

NH 2 b, CH b, CN str

1515

1498

1513

1516

Cyclopropanei

CH 2 str s

3027

3011

3058

3045

 

NH 2 b, CH b, CN str

1470

1463

1493

1493

 

CH 2 sciss

1499

1439

1532

1529

 

d CH 3

1450

1434

1527

1532

 

ring str s

1189

1186

1202

1202

 

CH 3 u, CN str

1429

1428

1433

1421

 

CH 2 twist

1127

1113

1156

1156

 

CH 3 u, CN str

1414

1401

1412

1412

 

CH 2 wagg

1067

1056

1091

1091

 

CH b, NH 2 b

     

CH 2 str as

3102

3070

3104

3104

 

CH b, CN str

1369

1369

1386

1386

 

CH 2 r

854

852

848

848

 

rings d, CN str, d CH 3

1345

1331

1353

1353

 

CH 2 str s

3019

3003

3044

3032

 

CH b, CN str

1327

1334

1350

1350

 

CH 2 str s

3019

3004

3045

3033

 

ring d, NH 2 r

1292

1309

1321

1323

 

CH 2 sciss

1440

1428

1452

1452

 

NH 2 r, CH b, CN str

1256

1246

1254

1251

 

CH 2 sciss

1440

1430

1453

1453

 

CH b, NH b

1232

1235

1246

1246

 

CH 2 wagg

1028

1018

1046

1046

 

CH b, NH 2 r, ring d

1199

1190

1203

1203

 

CH 2 wagg

1028

1024

1046

1046

 

d CH 3

1136

1121

1179

1179

 

CC str s

868

855

876

877

 

NH 2 r, d CH 3

1067

1049

1074

1074

 

CC str s

868

855

878

878

 

NH 2 r, d CH 3, CH b

1036

1034

1057

1057

 

CH 2 str as

3082

3047

3085

3085

 

NH 2 r, CH b

1000

970

988

988

 

CH 2 str as

3082

3048

3086

3086

 

CH oopb

958

949

958

957

 

CH 2 r

1191

1182

1198

1198

 

rings d, CH b, CH 3 r

895

894

904

904

 

CH 2 r

1191

1181

1199

1199

 

CH oopb

841

821

826

826

 

CH 2 tors

738

732

731

725

 

rings tors, CH oopb

800

792

803

803

 

CH 2 tors

738

738

736

727

 

rings d, CN str

730

736

736

737

      
 

rings d, d CH 3

715

718

727

727

Pyruvic acidj,k

OH str

3463

3411

3511

3509

 

rings tors

673

675

689

689

 

CH 3 str as

3025

3003

3036

3035

 

rings tors, CH oopb

640

644

656

656

 

CH 3 str s

2941

2925

2949

2925

 

NH 2 b, CC str, CN str

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 3 b as

1424

1417

1442

1441

GlcN-6Se

SO2 str s

1180

1082

1198

1200

 

CC str as

1391

1373

1387

1388

 

SO(H) str

885

775

859

861

 

CH 3 b s

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 3 r

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 3 r

1030

1009

1030

1030

       

OH tors

668

675

671

669

ßGlcNMe e)

NH\(_{3}^{+}\) str s

3242

3241

3314

3286

      
 

NH\(_{3}^{+}\) str as

3297

3287

3338

3338

Methyloxyranel

CH 2 str as

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 3 str as

2995

2957

2989

2988

 

OH(3) str

3612

3588

3659

3658

 

CH 3 str as

2974

2943

2979

2979

 

OH(6) str

3672

3672

3717

3717

 

CH 3 str s

2942

2929

2862

2959

GlyLysH +f

OH str

3584

3535

3609

3608

GlyGlyH +f

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

a) s=symmetric, as=asymmetric, str=stretching, r=rocking, b=bending, d=deformation, oopb=out of plane bending, oop=out of plane, tors=torsion. Experimental values taken from b) Ref. [15], c) Ref. [40], d) Ref. [44], e) Measured by us using the method given in our previous paper [37], f) Ref. [31], g) Ref. [27], h) Ref. [2], i) Ref. [26], j) Ref. [33], k) Ref. [6], l) Ref. [39]

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.

Fig. 2

Deviations of full B2PLYP, full B3LYP, and both hybrid MMFF94 calculations with respect to experimental values for dataset FS15

Assessment of the hybrid approach by comparison with experiment

All calculated frequencies are compared to the experimental ones of the both sets retrieved from the literature. Experimental frequencies are reported in Tables 1 and 2 for sets FS15 and FL13, respectively. Using these as reference, our statistical analysis includes: average unsigned error (AUE) and root-mean-square deviation (RMSD) and the maximal error (MaxUE) (reported in Table 3). Figure 2 shows the errors of our hybrid MMFF94 methods, full B2PLYP GVPT2 and full B3LYP GVPT2 calculations with respect to experiment for the set FS15. Hybrid MMFF94 GVPT2 method reproduces the experimental frequencies with a RMSD of 25 cm -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 2O2 (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 2O2. 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
Table 3

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

C2

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

N2

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

P2

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

H2CO

MaxUE

22

46

18

27

ßGlcNMe

MaxUE

59

72

47

 

AUE

10

25

13

15

 

AUE

17

49

44

 

RMSD

13

30

13

16

 

RMSD

26

50

44

H2O

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

NH3

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

SO2

MaxUE

36

120

40

40

DMSO4

MaxUE

117

34

33

 

AUE

25

71

25

25

 

AUE

91

16

15

 

RMSD

27

80

29

29

 

RMSD

96

20

19

H2O2

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

CH2NH

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

CH2O2

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

C2H4

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

H2O- H2O

MaxUE

34

64

26

63

     
 

AUE

12

34

13

23

     
 

RMSD

16

37

15

29

     

CH3OH

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.

Figure 3 shows the errors for the set FL13. The overall agreement of the hybrid MMFF94 GVPT2 anharmonic frequencies with the experimental ones is very good, with a RMSD ranging from 17 to 50 cm -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 ßGlcNMe 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.
Fig. 3

Deviations of full B3LYP and both hybrid MMFF94 calculations with respect to experimental values for dataset FL13

The statistical analysis of the deviations to the experimental frequencies is shown in Figs. 4 (set FS15) and 5 (both sets). Hybrid MMFF94 methods show a blue shift, while the full B3LYP approach shows a red shift. Full B2PLYP frequencies (small molecules only), are slightly blue shifted.
Fig. 4

Histogram of error analysis for FS15 set of molecules, using experimental frequencies as references. Negative values correspond to a red shift of calculated frequencies compared to experimental ones

Fig. 5

Histogram of error analysis for FS15 and FL13 sets of molecules, using experimental frequencies as references. Negative values correspond to a red shift of calculated frequencies compared to experimental ones

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.

The wavenumbers calculated in this study can be divided into two ranges. A first range of weak modes covering torsion, bending and stretching modes where hydrogen atoms are not involved. The frequencies for this range are below 2000 cm -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.
Fig. 6

Deviations of full B3LYP and hybrid MMFF94 calculations with respect to experimental values for FS15 and FL13. Statistic parameters are given for all frequencies, for low frequencies (below 2000 cm -1), and for high frequencies (above 2000 cm -1)

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

Finally, the total CPU time required for the calculations of the vibrational frequencies of larger molecules, with more than 30 atoms of our sets are reported in Table 4. The calculations were performed on the same computer for both the full B3LYP GVPT2 approach and the hybrid MMFF94 method.
Table 4

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.

For large-sized molecules, the main bottleneck of standard DFT GVPT2 calculation lies in the computation time of the third and fourth derivatives. For the larger molecule of our set (AVPO), the calculation of these derivatives is 321 times longer than the harmonic part of a GVPT2 calculation. This part is considerably reduced in our hybrid approach. The main bottleneck then becomes the GVPT2 calculation itself (see Fig. 7), where a diagonalization of a symmetric matrix is necessary. To overcome this limitation, we explored the accuracy of HDCPT2 method proposed in the literature as an alternative to the variational treatment of Fermi resonance. The results of this method, using the derivatives computed by our hybrid MMFF94 method, are given in Tables 1 and 2. These tables clearly show that HDCPT2 gives a very good accuracy, similar to that of GVPT2, while significantly reducing the computational time by a factor of 50 (see Table 4).
Fig. 7

On the left, the percentage of each step of the anharmonic calculation in the total CPU time for AVPO molecule. On the right, the total CPU time anharmonic calculation for each method using a logarithmic scale

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Loïc Barnes
    • 1
  • Baptiste Schindler
    • 1
  • Isabelle Compagnon
    • 1
    • 2
  • Abdul-Rahman Allouche
    • 1
  1. 1.Institut Lumière MatièreUMR5306 Université Lyon 1-CNRS, Université de LyonVilleurbanne CedexFrance
  2. 2.Institut Universitaire de France IUFParisFrance

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