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Department Chemie Lehrstuhl f ¨ ur Theoretische Chemie Technische Universit¨ at M ¨ unchen Theoretical Study of Anharmonic Vibrational Modes and Couplings with the VSCF Algorithm Mehdi Bounouar Vollst¨ andiger Abdruck der von der Fakult¨ at f¨ ur Chemie der Technische Universit¨ at unchen zur Erlangung der akademischen Grades eines Doktors der Naturwissenschaften genehmigten Dissertation. Vorsitzender: Univ.-Prof. Dr. F. E. K¨ uhn Pr¨ ufer der Dissertation: 1. Univ.-Prof. Dr. W. Domcke 2. Univ.-Prof. Dr. St. J. Glaser Die Dissertation wurde am 11.12.2007 bei der Technische Universit¨ at M ¨ unchen eingereicht und durch die Fakult¨ at f¨ ur Chemie am 13.02.2008 angenommen.

Tesis anharmónicos

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  • Department ChemieLehrstuhl fur Theoretische ChemieTechnische Universitat Munchen

    Theoretical Study of Anharmonic Vibrational Modesand Couplings with the VSCF Algorithm

    Mehdi Bounouar

    Vollstandiger Abdruck der von der Fakultat fur Chemie der Technische UniversitatMunchen zur Erlangung der akademischen Grades eines

    Doktors der Naturwissenschaften

    genehmigten Dissertation.

    Vorsitzender: Univ.-Prof. Dr. F. E. KuhnPrufer der Dissertation:

    1. Univ.-Prof. Dr. W. Domcke2. Univ.-Prof. Dr. St. J. Glaser

    Die Dissertation wurde am 11.12.2007 bei der Technische Universitat Muncheneingereicht und durch die Fakultat fur Chemie am 13.02.2008 angenommen.

  • Department ChemieLehrstuhl fur Theoretische ChemieTechnische Universitat Munchen

    Theoretical Study of Anharmonic Vibrational Modesand Couplings with the VSCF Algorithm

    Mehdi Bounouar

    A dissertation submitted to the Technische Universitat Munchen for the degree of

    Doktors der Naturwissenschaften

    [email protected]

  • Summary

    The rapid developments in laser technology and their use in multidimensional vi-brational infrared (IR) spectroscopy provide a powerful new tool to study, throughmolecular vibrations, the structure and dynamics of proteins and other biomolecules.With a temporal resolution down to the sub-picosecond regime, it becomes possibleto deepen our understanding of fundamental biochemical processes in the realm ofprotein folding and function.

    The direct connection between vibrational spectra, molecular structure, and inter-molecular interactions has made IR vibrational spectroscopy an established tool in thestudy of molecular matter. In particular for peptides and proteins, where the IR absorp-tion in the spectral range of 1400-1800 cm1 (amide-I band) has drawn attention formany years as a marker for the deduction of structural and dynamical information dueto its sensitivity to structure fluctuations. However, any interpretation obtained fromconventional linear IR absorption spectroscopy is at best qualitative due to inhomoge-neously broadened vibrational transitions in a narrow spectral region.

    The unprecedented potential of multidimensional vibrational IR spectroscopy to dis-entangle the congested vibrational spectra on a fast timescale have led to the need fortheoretical techniques applicable to vibrational states that are significantly perturbedfrom the harmonic oscillator limit. The complexity of the measured spectra in the caseof many vibrational degrees of freedom and the importance of anharmonic vibrationaleffects in many biological molecules is such that any quantitative interpretation dependin a crucial way on the comparison with reliable and accurate theoretical simulations.

    The Vibrational Self Consistent Field (VSCF) method with a hierarchical many-bodyexpansion of the potential energy surface (PES) can provide us with an efficient frame-work for the computation of the vibrational states of strongly anharmonic systems.Unfortunately the first principle computations of anharmonic vibrational states rapidlybecomes a daunting task with the increasing system size.

    In this thesis we try to address efficiently this computational bottleneck while achiev-ing reasonable agreement with experiment and understand the complex vibrationalcoupling network of the amide modes.

  • .

  • CONTENTS i

    Contents

    1 Introduction / Overview 1

    1.1 Multidimensional Vibrational IR Spectroscopy . . . . . . . . . . . . 1

    1.2 The Vibrational Problem . . . . . . . . . . . . . . . . . . . . . . . . 5

    1.2.1 The Harmonic Approximation . . . . . . . . . . . . . . . . . 6

    1.2.2 Second Order Vibrational Perturbation Theory . . . . . . . . 7

    1.2.3 The Vibrational Exciton Hamiltonian . . . . . . . . . . . . . 9

    1.2.4 Vibrational Self Consistent Field Theory . . . . . . . . . . . 11

    2 Vibrational SCF Theory 12

    2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    2.2 Hierarchical expansion of the PES . . . . . . . . . . . . . . . . . . . 14

    2.3 The Grid Problem in VSCF . . . . . . . . . . . . . . . . . . . . . . . 15

    2.3.1 Rectilinear Vs. Curvilinear Coordinates . . . . . . . . . . . . 15

    2.3.2 Grid Range . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    2.4 Correlation Correction . . . . . . . . . . . . . . . . . . . . . . . . . 18

    2.4.1 Vibrational Second Order Perturbation Theory . . . . . . . . 18

    2.4.2 Vibrational Configuration Interaction . . . . . . . . . . . . . 20

    2.4.3 Numerical Integral Rules . . . . . . . . . . . . . . . . . . . . 21

    2.5 Ab-Initio Potentials and VSCF . . . . . . . . . . . . . . . . . . . . . 25

  • CONTENTS ii

    2.5.1 Localization and Density Fitting Methods . . . . . . . . . . . 25

    2.5.2 Multi-Level Calculations, Model Potentials andCoupling Norms . . . . . . . . . . . . . . . . . . . . . . . . 26

    2.6 VSCF Implementation and Benchmarking . . . . . . . . . . . . . . . 31

    2.6.1 Formaldehyde . . . . . . . . . . . . . . . . . . . . . . . . . 31

    3 Formamide and Thioformamide 38

    3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

    3.2 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . 39

    3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 40

    3.3.1 Formamide . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

    3.3.2 Thioformamide . . . . . . . . . . . . . . . . . . . . . . . . . 51

    3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

    4 N-methylacetamide as a Model for Peptide Linkage 58

    4.1 N-Methylacetamide . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

    4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

    4.1.2 Computational Methods . . . . . . . . . . . . . . . . . . . . 60

    4.1.3 Vibrational Spectrum . . . . . . . . . . . . . . . . . . . . . . 61

    4.1.4 Multi-Level Calculations . . . . . . . . . . . . . . . . . . . . 65

    4.1.5 Coupling Potential Norms and Global Selection Schemes . . . 66

    4.1.6 Advanced Mode Coupling Selection Schemes . . . . . . . . . 74

    4.1.7 Amide Modes and Anharmonicity . . . . . . . . . . . . . . . 78

    4.1.7.1 Rectilinear Displacements . . . . . . . . . . . . . . 79

    4.1.7.2 Curvilinear Displacements . . . . . . . . . . . . . 83

    4.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

  • CONTENTS iii

    5 First-Principle Study of Hydrogen Bonded Complexes 90

    5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

    5.2 Hybrid DFT Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 92

    5.3 Multi-Coefficient Correlation DFT Methods . . . . . . . . . . . . . . 93

    5.4 Spin Component Scaling . . . . . . . . . . . . . . . . . . . . . . . . 93

    5.5 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . 94

    5.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

    6 Conclusions and Future Work 101

    A Additional Tables and Figures 103

    B Collocation with Distributed Gaussian Basis Set 108

    References 110

    List of Figures 129

    List of Tables 131

    Acknowledgements 133

  • 1Chapter 1

    Introduction / Overview

    1.1 Multidimensional Vibrational IR Spectroscopy

    Investigating the three-dimensional structure of proteins in general and the foldingevents in particular is of prime importance for understanding their biological func-tion. Protein folding is one of the key processes in living organisms as a particularstructure determines its specific activity, and any biological function is supported by acorresponding protein system [1]. Folding from the amino acid chain into the correctthree-dimensional structure involves reactions that span from the picosecond rotationof the molecular bonds and rearrangements, to external forces over the formation ofsecondary structures in the nanosecond and microsecond range, to the correct arrange-ment reached on the time scale of seconds [2]. Non-bonded interactions, mainly hydro-gen bonds, play a significant role and in many cases the equilibrium fluctuations aroundthe time-averaged structure are essential to the proteins functionality [3]. Therefore,a short experimental time-scale is essential in order to probe structures which dependon interactions that constantly rearrange, dissociate and reform under ambient condi-tions [4].Although in the past years, traditional X-ray diffraction and nuclear magnetic res-

    onance (NMR) spectroscopy provided (and still provides) a wealth of structural in-formation [5], they have known limitations. While the former can only give a staticpicture corresponding to a solid-state structure1 (therefore, certain classes of proteins

    1 This is partially true, small-angle X-ray scattering (SAXS) techniques can be used for the struc-tural determination of proteins in solutions [6] but then only with a low-resolution (although, severalimprovements have been made lately in resolution and reliability [7]).

  • 1.1 Multidimensional Vibrational IR Spectroscopy 2

    are difficult to investigate due to crystallization problems), the latter, can only measureprocesses that takes place on a microsecond to millisecond time scale. Any detailedinformation on the fast protein dynamics, is therefore lost.

    The advent of picosecond and femtosecond laser technologies and the rapidly grow-ing possibilities to control optical and IR pulses on those time scales have allowedthe recent successes in the experimental realization of coherent multi-dimensional IRspectroscopy. Particularly, it opened the exciting possibility to study the molecularstructure and dynamics of polyatomic systems in solution on timescales 6 to 10 or-ders of magnitude faster than NMR spectroscopy. With a temporal resolution down tothe sub-picosecond regime [8, 9, 10], multi-dimensional IR spectroscopy techniqueshave the unique potential to unveil the structure and fast conformational dynamics ofproteins and other biomolecules, by directly following the elementary events on smallmodel systems [11] and biologically relevant systems [12, 13] in great details.

    1500 1600 1700

    3200 3300 34002800 2900 3000 3100

    800 1000 1200 1400

    CH as3

    CH ss3

    C H s

    Amide B

    Amide A

    Amide I

    Amide II

    Figure 1.1: IR spectra in the mid-infrared region (800 -3400cm1) of a ten-residue -poly(L)-alanine , with theelectric vector perpendicular (full line) or parallel(dashed line) to the direction of orientation, takenfrom Ref. [14]

    The IR absorption spectra of peptides and proteins are dominated by vibrationalbands that can be described in terms of oscillators localized in each repetitive unitand their mutual couplings [15]. These spectroscopically well resolved bands presentin the 3000 -3500 cm1 and 1500 -1800 cm1 region, are denoted amide bands (seeFigure 1.1). They have been studied extensively for many years because of their strong

  • 1.1 Multidimensional Vibrational IR Spectroscopy 3

    dependence on their local environment and structural patterns. The amide-I vibrationalmode (found at 1500 -1800 cm1) which involves mainly the C=O stretch coordinatewith contributions from the C-N stretch and N-H bend motions (see Figure 1.2) is, atleast from an experimental point of view, the most important one due to its large tran-sition dipole moment and sensitivity to secondary structure. Other important bandsare the amide-A/B (3000 -3500 cm1, essentially N-H stretch) and amide-II (1500 -1550 cm1, mainly in-plane N-H bending) .Conventional one-dimensional linear IR absorption experiments yield broad unre-

    solved bands, resulting from many inhomogeneously broadened vibrational transitionsin a narrow spectral region. The information provided by the characteristic amide-Iband line-shapes that corresponds to the different secondary structural elements en-countered in proteins [16, 17], is in general only sufficient to qualitatively or semi-quantitatively assign a relative ratio of secondary structure motifs (-helix vs. -sheet)contained in a particular protein2.

    Figure 1.2: Amide-I, -II and -A vibrations of a peptide group

    Multidimensional IR spectroscopy on the other hand has the potential to disentanglesuch highly congested vibrational spectra of biomolecules originating from various ef-fects such as conformational fluctuations and their coupling to the local environment.In nonlinear 2D-IR spectroscopy the structural and dynamical information is typicallypresent in terms of diagonal- and cross-peak locations, shapes and intensities and theirrespective temporal evolution. Spreading the spectral information along independentfrequency axes allows us to see correlations and anharmonic couplings between vari-

    2 Proteins with predominant -helical structure exhibit amide-I and amide-II absorptions in the spec-tral range 1652 -1657 cm1 and 1545 -1551 cm1, respectively, in aqueous solution, while proteins withpredominantly -sheet structure exhibit similar contributions both at low 1630 -1640 cm1 as well ashigh frequencies (above 1680 cm1) and 1521 -1525 cm1.

  • 1.1 Multidimensional Vibrational IR Spectroscopy 4

    ous vibrational modes, to some extent in a similar way to multidimensional NMR spec-troscopy [14, 18, 19, 20, 21], but on a significantly faster timescale. The comparison isrestricted even more by the fact that the vibrational excitation of a polyatomic systemby the absorption of a photon initiates a complex sequence of dynamical processes.The Hamiltonian that governs the underlying physics is complex and the analysis ofrecorded experimental spectra of large molecules based on simple analogies deducedfrom model systems is difficult.

    The simplest spectroscopic models are actually based on the assumption, that eachindividual peptide site carries a localized amide-I oscillator that couples electrostati-cally to the next neighboring sites, neglecting couplings to other vibrational modes.Although this type of modelling can give qualitative agreement with the experimentalresults, it is based on the assumption of a separation of a set of spectroscopically rel-evant vibrations. This is a very restrictive approach, while in reality the mechanismsof vibrational energy transport in peptides are still not well understood. More com-plex vibrational couplings can take place and anharmonicity plays a pivotal role in aproteins vibrational dynamics and energy transfer [22].In general, vibrational couplings are classified as either occurring through-space or

    through-bond [21, 23, 24, 25, 26, 27]. Through-space interactions are typically elec-trostatic in nature and have been successfully described using multipole or even dipole-dipole interaction models [15, 28, 29]. In comparison, the through-bond interactionsare typically short ranged and need to be described employing ab-initio quantum chem-ical (QC) methods [23, 30].In this context, a first principle detailed analysis of the anharmonic couplings present

    within a single peptide plane is called for in view of the wide-spread use of vibrationalmodel Hamiltonians based on the assumption that the amide-I vibrational modes areseparated from the remaining protein modes. This is of interest not only for the amidevibrational modes, but also for any other potential marker that may be useful forprobing different facets of molecular structure and properties. Recently, the C-H

    vibration has been suggested as a secondary structure sensitive probe [31], showingthat a detailed study of CH vibrations in oligo-peptides is also necessary along with themore commonly studied modes. However, in the case of a C-H vibrational mode,any model based on the assumption of a localized oscillator is unlikely to correctlydescribe such a strongly coupled and delocalized vibrational mode.

  • 1.2 The Vibrational Problem 5

    1.2 The Vibrational Problem

    As discussed in the previous section, the first principles prediction of anharmonic vi-brational energy levels for polyatomic molecules is an important goal in theoreticalchemistry, it should not be regarded as an unnecessary complication, but rather as anessential condition to understand the basic biological phenomena.

    We start from the non-rotating (J = 0) quantum-mechanical vibration-rotation Hamil-tonian3 (Watson Hamiltonian) [32, 33] in atomic units (a.u.) with mass-scaled normalmode coordinates in the Born-Oppenheimer (BO) approximation

    HW =12

    ,pi pi

    Tc

    12 i

    2Q2i

    T

    18

    Tw

    +V (Q) (1.1)

    where the subscripts and refer to the x, y and z components of the cartesiancoordinates. The inverse effective moment of inertia tensor is denoted , and pi isthe cartesian component of the vibrational angular momentum given by

    pi = i< j

    ()i j(QiPj Q jPi) (1.2)

    where Pi is the momentum conjugate to the normal coordinates Qi and ()i j is theCoriolis coupling constant. As we are primarily interested in studying large molecules,the contributions of the vibration-rotation Coriolis interaction (Tc) as well as the massdependent terms (Tw, also called Watson term) to the vibrational Hamiltonian are cer-tainly negligible in our conclusions as both contributions scale with the inverse of themoments of inertia tensor of the whole molecule. The approximations in the treat-ment of the potential expansion and the choice of quantum mechanical methods forthe computation of the potential energy surface (PES) typically have a larger effect onthe vibrational spectrum than the neglected ro-vibrational corrections. Our vibrationalHamiltonian then takes the more familiar form

    H =12 i

    2Q2i

    +V (Q) (1.3)

    3We use here (Q) as a shorthand notation for (Q1, ...,QN)

  • 1.2 The Vibrational Problem 6

    So far, even if we have chosen here to use normal mode coordinates Qi to formulateour Hamiltonian in Eq. (1.3), no assumption or approximation has been made on thepotential V (Q) which is here supposed to be exact.

    1.2.1 The Harmonic Approximation

    One of the simplest approximation that one can make about the potential V , is to as-sume a harmonic form. If we make the assumption of infinitesimal displacements ofthe nuclei from their equilibrium positions, we can expand the potential in a Taylorseries for a set of general coordinates qi :

    V (q) = V0 +N

    i=1

    (Vqi

    )0

    qi +12

    N

    i, j=1

    ( 2Vqiq j

    )0

    qiq j + (1.4)

    If we are at a global minimum of the PES and neglect terms higher than the secondorder, the equation above is greatly simplified [34, 35]. Only the first and third termin the l.h.s of Eq. (1.4) remains and the solution of the vibrational problem becomesstraightforward. This normal mode analysis is usually carried by first the evaluation,either analytically or numerically (by performing small atomic displacements from thereference geometry), the elements of the force constant matrix

    Fi j =( 2V

    qiq j

    )0

    (1.5)

    where now the qis are cartesian coordinates. This force constant matrix is then sub-sequently transformed into mass-weighted cartesian coordinates

    F(M) =M

    12F M

    12 (1.6)

    whereM is a diagonal matrix with nuclear mass elements mi. Solving this eigenvalueproblem, we obtain after proper unit conversion4, the frequencies in wavenumber unitsand the corresponding normal mode coordinate vectors Qi.

    4if i and Li are respectively eigenvalue and eigenvector of F(M) in [a.u] , multiplying 12i by a

    conversion factor of ca. 5140.48733 returns the frequency in cm1. Normal coordinates are obtainedwith Qi =M 12 Li .

  • 1.2 The Vibrational Problem 7

    Nowadays, the prediction of vibrational energies at the harmonic level is done rou-tinely by most quantum chemistry packages. Nevertheless, the vibrational harmonicoscillator is a highly idealized system, where the eigenstate energy of a single vibra-tional mode is

    (ni)i = ~i

    (ni +

    12

    )(1.7)

    and the corresponding harmonic wavefunction

    (ni)i (Qi) = Nni Hni (iQi)e(2i Q2i /2) , i =

    (i~

    )1/2(1.8)

    where Hni is a Hermite polynomial and Nni a normalizing factor (see Refs. [36] and[35]). In this approximation, the fundamental frequency i is independent of the ampli-tude, irrespective of the amount of energy injected into the system and only the transi-tions that satisfy the vibrational selection rule n1 are allowed5. Consequently, over-tones and combination bands are forbidden [35], resulting in a physical picture whichis inconsistent with what is commonly seen experimentally for polyatomic molecules.

    1.2.2 Second Order Vibrational Perturbation Theory

    The theory of the quantum-mechanical anharmonic oscillator using a second orderperturbative treatment and the potential expansion of Eq. (1.4) up to quartic force con-stants has been known for some time now [34, 35, 37, 38]. Nevertheless, it is onlyrecently that anharmonic vibrational frequency computations using second order per-turbation theory (PT2), have been implemented by Barone [39, 40] in the popularGaussian [41] QC package.The total vibrational energy of a molecule is expanded in terms of powers of vibra-

    5 In the harmonic dipole approximation the electric dipole transition is written as(n)

    (m)= ( Qi)

    0

    (ni)i

    Qi(mi)i and from the properties of the Hermite polynomials, it follows that only transitions mi = ni 1 arepossibly non-zero.

  • 1.2 The Vibrational Problem 8

    tional quantum numbers ni and can be written as follows

    E = E0 +~N

    i

    i

    (ni +

    12

    )+~

    N

    i j

    xi j(

    ni +12

    )(n j +

    12

    )+ (1.9)

    where i is the i-th harmonic frequency. The anharmonic constants xi j can be ob-tained from

    2xii =18

    [iiii +

    52iii3i

    + k 6=i

    2iik(82i 32k

    )k(42i 2k

    ) ] (1.10)

    xi j =14

    (ii j j +

    iiii j ji

    + j j jii j

    j+

    22ii ji42i 2j

    22j ji j

    42j 2i

    )

    +14 k 6= j 6=i

    iik j jk

    k

    22i jk(

    2i +2j +

    2k

    )k

    4i +4j +

    4k 2

    (2i

    2j +

    2i

    2k +

    2j

    2k

    ) (1.11)

    where the cubic i jk and quartic i jkk force field constants in Eqs. (1.10) and (1.11) arethe third and fourth derivatives of the PES with respect to the normal mode coordinates.These force constants are obtained by numerical central differentiation of the analyticalsecond derivatives at geometries displaced by small increments from the referencegeometry.

    When analytical second derivatives are implemented the perturbative approach canbe very appealing, especially if efficiently parallelized implementations are becomingavailable. Although an important tool, there are situations where this approach in itssimplest form fails. Unfortunately the formamide amide group out of plane motionfalls into such a category [40]. Although an integrated perturbative-variational ap-proach has been suggested in order to correct this [40], the method inherently onlyincludes anharmonic effects resulting from small displacement w.r.t. the equilibriumgeometry. Thus, while PT2 is certainly a method of choice for semi-rigid molecules,it should be used with care for highly anharmonic and floppy molecules like peptidesand proteins.

    In a three-pulse photon-echo 2D-IR spectrum as shown in the schematic in Fig-ure 1.3 (left) for two coupled oscillators i and j, there are five peaks for each oscilla-tor. The diagonal peaks labelled (a) and (b) are separated by the diagonal anharmonic-ity i, while the cross-peaks labelled (c) and (d), are separated by the off-diagonal

  • 1.2 The Vibrational Problem 9

    Figure 1.3: (Left) 2D-IR spectrum of two coupled oscillators.(Right) Two mode vibrational level diagram. Figuresadapted from Refs. [8] and [42].

    anharmonicity i j. Hence the peak separation in a 2D-IR spectrum provides us with adirect measure of the anharmonicity. For proteins, peptides, and peptide model systemsthe amide-I modes diagonal anharmonicities i typically range from 10 to 20 cm1 andup to ca. 160 cm1 for amide-A vibrational modes. The mixed-mode anharmonicityi j of neighboring amide-I modes of an -helix is approximatively 9 cm1 [43]. Allthe difficulty now, is in obtaining these anharmonic constants from first principle calcu-lations. Assuming that in Eq. (1.9) all terms higher than bilinear interaction are negligi-ble, such a two mode system can also be represented by a more familiar level diagram,as shown in Figure 1.3 (right), where i =2xii and i j =xi j. Using a PT2 approachthe anharmonic constants can be obtained via Eqs. (1.10) and (1.11). However, if weare provided with a method that allows us to compute directly the vibrational energyof fundamentals, overtones and combination bands in the normal mode basis (as wewill see later in the next Chapter), we can also deduce the anharmonic constants fromthe energy states as shown in Figure 1.3 (right).

    1.2.3 The Vibrational Exciton Hamiltonian

    The structural determination of any biologically relevant system through vibrationalspectroscopic techniques implies studying models of at least thousands of atoms in

  • 1.2 The Vibrational Problem 10

    their natural environment (water in most cases). It is therefore a euphemism to say thata detailed ab-initio anharmonic treatment is unlikely to be tractable in the near futurefor such systems. For this reason a more intuitive and applicable (to large systems)concept has to be used.

    When excited, a vibrational mode like for instance amide-I, interacts either throughspace or through bond with neighboring amide units. Separating the amide-I vibra-tional manifold of states from all other vibrational degrees of freedom, this excitationcan move from site to site, thus giving rise to a delocalized exciton-state i.e. a vi-brational Frenkel exciton [44]. A general vibrational Hamiltonian in terms of Bosoncreation and annihilation operators can be written in the following way

    H(t) =N

    i

    iBi Bi +N

    i

    i2

    Bi Bi BiBi

    +N

    i 6= j

    Ji jBi B j +N

    i 6= j

    Ki j2

    Bi BjBiB j (t)

    (1.12)

    where Bi and Bi are the creation and annihilation operators for the ith mode, withfrequency i and diagonal anharmonicity i. These operators satisfy the Boson com-mutation relation

    [Bi,B

    j]= i j. The harmonic intermode coupling is designated by Ji j.

    In other words, an amide-I spectrum for proteins can be obtained by diagonalizing aHamiltonian constructed in the basis of peptide units. The parameters necessary forthis Hamiltonian can be obtained from experiment or from ab-initio computations. Inthe simplest models, the amide-I coupling elements (i 6= j) are in general calculatedfrom transition dipole interactions between oscillators located in the peptide units [23,28, 29].Several models and computational strategies that generally aim at representing either

    only the amide-I or a slightly larger vibrational subspace of the vibrational modes ofthe protein have been developed over the past years [19, 28, 45, 46, 47, 48, 49, 50, 51]and have proven to be quite effective.

    Although this type of modelling can give qualitative agreement with the experimentalresults, it should be given a firmer basis by approaches involving detailed theories ofanharmonicity and high-resolution experiments on peptides of increasing sizes.

  • 1.2 The Vibrational Problem 11

    1.2.4 Vibrational Self Consistent Field Theory

    The Vibrational Self Consistent Field (VSCF) method, which can yield approximatefrequencies and wavefunctions of an anharmonic vibrational system, borrows manyconcepts from electronic structure theory and maps them to the anharmonic vibra-tional coupling problem. Surprisingly though, its development started only in the lasttwo decades, predominantly with the work of Bowman, Carter and Gerber [52, 53, 54](see also references in Ref. [55]). VSCF could be one of the most effective tools tostudy anharmonic vibrational states of large polyatomic systems. However, severalapproximations and improvements are still needed for more general and reliable appli-cation. The theory is presented in more detail in the next chapter.

  • 12

    Chapter 2

    Vibrational SCF Theory

    2.1 Introduction

    Despite the rapid advances in hardware and software technology, the computationaleffort required to study the properties of proteins or even peptides with spectroscopicaccuracy is still overwhelming. Most theoretical approaches for the direct computationof accurate vibrational states of coupled anharmonic vibrational modes are inherentlyrestricted to small systems with only a a few degrees of freedom.

    Methods based on perturbation theory (Section 1.2.2) are not appropriate for moleculeswith large amplitude motions and strong mode coupling such as peptides and proteinsand do not perform well far from equilibrium. Other methods based on localized modeHamiltonians (see Section 1.2.3), although applicable to real-life systems, heavily de-pend on parameters and approximations that have not provided any deeper insight inthe validity of the approximations that enter the model nor its reliability.

    No currently available theoretical method fulfils all requirements needed to success-fully describe the vibrational multidimensional spectra of biomolecules, i.e. yield ac-curate vibrational force constants beyond the harmonic approximation for a wide rangeof bond types with a correct description of non-bonded interactions, perform well closeto the equilibrium structure as well as in non-equilibrium situations and last but notleast have a moderate (optimally linear) scaling with respect to system size.The VSCF theory in which each vibrational mode is described as moving in an ef-

    fective field, offers the possibility to overcome some of these limitations. It can yieldapproximate frequencies and wavefunctions for the anharmonic vibrational system that

  • 2.1 Introduction 13

    can also be used to simulate multi-dimensional IR spectra by sum-over-states or prop-agation techniques [56]. Furthermore, because of the close relationship of the VSCF tothe MCTDH method [57] any PES model that is computationally well suited for VSCFwill also be usable in the MCTDH context for the computation of time-dependent mul-tidimensional spectra. Recently, Franck-Condon integrals of polyatomic molecules,which are useful in spectral simulations of various kinds have also been computed onthe basis VSCF anharmonic vibrational wave functions [58, 59].Starting from the description of the vibrational wave function in terms of a Hartree

    product of N effective one-dimensional anharmonic vibrational wave functions

    (n) (Q) =N

    i=1

    (ni)i (Qi) (2.1)

    and using the Hamiltonian of Eq. (1.3), we can write the vibrational Schrodingerequation in terms of mass-weighted normal mode coordinates Qi as follows1[

    12

    N

    i=1

    2Q2i

    +V (Q)]

    (n) (Q) = E(n)(n) (Q) (2.2)

    Applying the variational principle

    (n) (Q) H (n) (Q)

    (n) (Q) |(n) (Q) = 0 (2.3)

    yields a set of N effective single-mode VSCF equations[

    12

    2Q2i

    + Vn,i (Qi)]

    (ni)i (Qi) = (ni)i (ni)i (Qi) (2.4)

    The effective potential Vn,i (Qi) for the mode Qi is given by

    Vn,i (Qi) =

    N

    l 6=i

    (nl)l (Ql)V (Q) N

    l 6=i(nl)l (Ql)

    (2.5)

    where the integration is extended over all but the i-th coordinate. The resulting set ofVSCF equations has to be solved iteratively until self-consistency is achieved.

    1We use here (Q) and n respectively as a shorthand notation for (Q1, ...,QN) and for a given stateconfiguration

    n1, ,nN

  • 2.2 Hierarchical expansion of the PES 14

    Using Eqs. (2.1), (2.4) and (2.5) we can write the total energy in the VSCF approxi-mation as

    E(n)V SCF =N

    i

    (ni)i +(1N)

    N

    i

    (ni)i (Qi)V (Q) N

    i(ni)i (Qi)

    (2.6)

    The similarity with electronic SCF theory is obvious. We can easily recognize theequivalent (in the Hartree theory sense) zeroth-(VMP0) and first order (VMP1) ener-gies, respectively first and second part in Eq. (2.6).Choosing an appropriate one-dimensional basis set (i) for each vibrational motion is

    primordial, as they define the building blocks of the anharmonic vibrational wave func-tion in Eq. (2.1) and will therefore determine its quality (flexibility and compactness).The basis set can be chosen to be harmonic oscillators or can be obtained by solvingthe one-dimensional Schrodinger equations through the so called discrete variable rep-resentation (DVR) [60], Fourier grid Hamiltonian method (FGH) [61] or collocationwith a distributed Gaussian basis set [62, 63, 64, 65] as was done in this work (see alsoAppendix B).

    2.2 Hierarchical expansion of the PES

    The evaluation of the multidimensional integrals involving the potential of Eq. (2.5)poses the main computational difficulty for large systems, and as first suggested byCarter [66] can be approximated by expanding the PES in terms of a hierarchical ex-pansion

    V (Q) =N

    i

    V (1)i (Qi)+N

    i< j

    V (2)i j (Qi,Q j)+N

    i< j

  • 2.3 The Grid Problem in VSCF 15

    unfavorably as n=1(N

    n

    )(Ngrid

    )n, where Ngrid is the grid size. For large molecules, it is

    therefore necessary to approximate or neglect several interaction potentials for n 2.

    2.3 The Grid Problem in VSCF

    2.3.1 Rectilinear Vs. Curvilinear Coordinates

    The separability approximation Ansatz made in Eq. (2.1) already suggested that theaccuracy of VSCF depends strongly on the choice of coordinates [67]. Indeed, for themethod to succeed, appropriate coordinates should incorporate strong interaction termsbetween atoms already at the single-mode displacement level. Additionally, collectivedisplacement should (optimally) minimize the interaction between modes in the PESmany-body expansion approximation. Ideally, pair-terms V (2)i j (Qi,Q j), should alreadybe negligible and not involve any near resonances between modes.

    Rectilinear normal mode coordinates are a common choice for Qi as they usually de-scribe the fundamental transitions of low-lying vibrationally excited states reasonablywell [34, 35, 38] and provide already a weakly coupled basis. Unfortunately, they areinherently not adapted to describe large amplitudes of motions, as we can see from Fig-ure 2.1, for the PES of a methyl deformation (rotational) mode of N-methylacetamide(NMA) on a dimensionless grid. Here, the dimensionless normal mode coordinates Qiare defined as

    Qi =(ii~

    ) 12 Qi (2.8)

    where i is the reduced mass of the i-th mode, and i its frequency. An abscissa valuexi in Figure 2.1 therefore corresponds to a displacement from the equilibrium geometry

    of xi (

    ~

    ii

    ) 12 Qi. As we can see, the description of torsional modes in cartesian normal

    coordinates like methyl deformation is especially difficult. Typically, rotational energybarriers for methyl groups without major sterical hindrance, are on the order of 0.002Hartree. At large amplitudes, energies are dominated by the CH bond stretch, conse-quently excited states can be difficult to converge in VSCF because couplings betweenlow (torsions) and high (stretches) frequency modes are increased artificially.A more appropriate coordinate system that allows for a better separability, as well as

  • 2.3 The Grid Problem in VSCF 16

    -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70Methyl deformation Normal Mode Coordinates

    0

    0.01

    0.02

    0.03

    0.04

    0.05

    0.06

    Ener

    gy [a

    .u]

    C

    H

    H H

    C

    H

    H H

    H

    H

    H

    Figure 2.1: Diagonal potential for a methyl torsional mode ofNMA with displacements shown at the equilibrium andthe maximally deformed geometry

    a more appropriate representation of the PES for large amplitude motions, would beinternal coordinates. Unfortunately, using such a representation has a price. Accordingto Podolsky and the latter reformulation by Schaad et al. [68, 69], the Hamiltonian ingeneralized mass-weighted coordinates can be written as follows

    H =12 i, j g

    1/2 pig1/2gi j p j +V (q) (2.9)

    where pi, p j are momentum operators, g is the kinetic energy coupling tensor, gi j

    denotes the elements of the inverse of g and the vector q is a set of internal coordi-nates. The problem here is the determination of the kinetic energy coupling tensorwhich rapidly becomes cumbersome as there are no simply derivable expressions inthe case of polyatomic molecules. Using a normal mode representation in cartesiancoordinates in order to uncouple (diagonalize) the kinetic energy operator to obtainEq. (2.2), is therefore difficult to avoid for large polyatomic molecules at the presenttime. However, we will see in Chapter 3 that, at least for the fundamental frequenciesand low lying excited states, using a normal mode representation in internal coordi-nates with the assumption that these coordinates uncouple the kinetic energy operator,is a crude approximation that works reasonably well.

  • 2.3 The Grid Problem in VSCF 17

    2.3.2 Grid Range

    In order to achieve a certain numerical stability, it is obvious that in a grid representa-tion of the PES, we have to choose suitable modal grid ranges. Here, by grid range wemean the maximally allowed displacements along a mode vector Qi. In other words,we generate the set of coordinates {Xi } necessary for the grid representation of adiagonal potential for the i-th modes as the following

    Xi = X0 +iQi, i =(

    ~

    ii

    ) 12, [ai,bi] (2.10)

    where X0 is the geometry at the equilibrium and Qi the normal mode vector. Withoutany a-priori knowledge of the potential, choosing an appropriate range is not a sim-ple task. A common choice that works reasonably well for many cases is to choose, [4,+4]. However, when the second order derivative of the PES near the min-imum becomes sufficiently small (low frequency modes), the induced displacementscan become important, as previously shown in Figure 2.1. Thus, grids should prefer-ably be adapted for every single mode. They should be sufficiently large to allow thevibrational wavefunction to asymptotically go toward zero and become balanced, butonly sufficiently so to avoid as much as possible convergence problems and other dif-ficulties that might occur when the resulting atomic displacements are becoming tooimportant.

    In this work, we obtained mode-optimal grids by first evaluating the single mode po-tentials at the Hartree-Fock (HF) level employing a double or triple zeta basis set withfew points on a sufficiently large grid. After interpolation on a denser grid, an appro-priate range can be selected for every mode based on the vibrational wavefunction. Aswe can see from Figure 2.2, using less expensive methods like the semi-empirical PM3or minimal basis sets is generally not a good choice. For low frequency modes (like inFigure 2.2), the PM3 potentials are often too far from the second-order MllerPlessetperturbation theory (MP2) potentials to be used as reasonable guess. It is of coursepossible to devise a more refined algorithm that would enable us to generate optimizedgrids on the fly, but in our case the straightforward approach is more efficient becauseit allows for a better parallelization.

  • 2.4 Correlation Correction 18

    Figure 2.2: Diagonal potentials (continuous) and their respec-tive wave function (discontinuous) for a low frequencymode of NMA (see Table 4.1 mode Nr.3) computed atdifferent levels

    2.4 Correlation Correction

    Analogously to the mean field approach in electronic structure theory which ignoresthe correlated motion of electrons, the VSCF method introduces a correlation er-ror due to the effective nature of the interactions between the oscillators. Again,following electronic structure theory, it is possible to correct this by including vi-brational correlation effects in VSCF using either second order perturbation theory(VMP2) [70], coupled-cluster (VCC) [71, 72], configuration-interaction (VCI) [73,74] or a multi-configurational SCF (VMCSCF) and its extension complete-active-space SCF (VCASSCF) [75, 76]. All these methods have of course their respectiveadvantages and drawbacks, however in this work we will only focus on the VMP2 andthe VCI approach.

    2.4.1 Vibrational Second Order Perturbation Theory

    With respect to its simplicity and computational cost it is rather surprising that thevibrational MllerPlesset perturbation theory [70] approach was introduced only sev-eral years after the VCI methods [73, 74]. The MP2 correction to the VSCF energy

  • 2.4 Correlation Correction 19

    can be written as

    E(n)MP2 = m6=n

    N

    l 6= j

    (ml)l (Ql)V

    N

    l 6= j

    (nl)l (Ql)

    2

    E(n)0 E(m)0

    (2.11)

    where the perturbation potential V is defined as

    V = V (Q)N

    j=1

    Vn, j(Q j) (2.12)

    E(n)0 and E(m)0 are respectively the zeroth-oder energies (see Eq. (2.6)) in the reference

    and virtual excited state.

    From the denominator of Eq. (2.11) it is clear that the perturbation approach in thecase of degenerate or nearly degenerate states, which is unfortunately ubiquitous for amanifold of vibrational states, will generate numerical instability. By expressing theMP2 correction differently

    E(n)MP2-m =

    12

    m6=n

    [E(m)0 E

    (n)0

    ((E(n)0 E

    (m)0

    )2+4

    N

    l 6= j

    (ml)l (Ql)V

    N

    l 6= j

    (nl)l (Ql)

    2) 12]

    (2.13)

    where the sign in Eq. (2.13) is negative when En < Em and vice-versa, a significantimprovement in the numerical stability is obtained. This expression is very similar tothe one that has been recently published in Ref. [77]. It is correct in the limiting caseswhere the numerator of Eq. (2.13) tends to zero and avoids the numerical instabilitywhen near degeneracy occurs.

    This equation has also the advantage that it can be evaluated alongside the regu-lar MP2 correction with almost no computational overhead allowing comparison withthe results from the canonical MP2 correction which gives a strong indication of theaccuracy of a second order perturbation treatment (see for example Table 2.1). A per-turbation theory formulation for the degenerate case [78] is of course possible, but asthe method does not offer a systematic and detailed approach comparable to VCI the

  • 2.4 Correlation Correction 20

    latter is generally preferable whenever it is tractable. In order to distinguish the regularVMP2 (Eq. (2.11)) from the modified version (Eq. (2.13)) we will refer to the latter asVMP2-m.

    2.4.2 Vibrational Configuration Interaction

    When two or more strongly mixed states of a coupled oscillator are accidentally de-generate or nearly degenerate giving rise to resonance, this becomes an obvious reasonfor the breakdown of the VSCF method as well as its perturbative extension. For suchcases that cannot be described using a single Hartree-Product, the VCI approach canbe a very accurate method and, as shown by Truhlar [79], if only specifically chosenstates are introduced in the VCI expansion the space-size can be controlled withouttrading accuracy.

    In the VCI algorithm the eigenfunctions of the vibrational Schrodinger equation areexpanded in a basis of VSCF states [73, 74] as follows

    (n)CI = i

    c(n)i

    (N

    k=1

    (n)k (Qk))

    (2.14)

    where (n)k (Qk) are eigenstates of the VSCF Hamiltonian. The eigenenergies E(n) aresolutions of the generalized eigenvalue problem

    det [HES] = 0 (2.15)

    where the elements of H, the full vibrational Hamiltonian, and S, the overlap integralmatrices are

    Hi, j =

    N

    i

    (ni)i (Qi) H N

    j(n j)j (Q j)

    (2.16)

    Si, j =

    N

    i

    (ni)i (Qi) N

    j(n j)j (Q j)

    (2.17)

    Here, the tilde symbol [ ] is used to differentiate modal bases out of two differentVSCF computations i and j. Looking at Eq. (2.17) it becomes clear that if we aregiven an orthogonal basis set the VCI approach is greatly simplified.

  • 2.4 Correlation Correction 21

    In a regular VCI expansion, different VSCF states are not orthogonal, but using avirtual basis set of vibrational configurations obtained from the wave function of aprevious VSCF effective Hamiltonian of a particular state (generally the ground state),the variational theorem applies to the resulting spectrum of eigenvalues and S becomesthe unit matrix because of orthogonality. This VCI approach using virtual basis sets orvirtual VCI is particularly useful. From a numerical point of view, converging multiplyand highly excited states, as would normally be required in order to build the Hamil-tonian of Eq. (2.16), is difficult to achieve. As we are interested only in a few specificstates and not the whole spectrum the virtual VCI approach is therefore preferable.

    As mentioned previously in Ref. [79], if only a few specific configurations are in-troduced in the VCI expansion space, it is possible to obtain, in an computationallyefficient way, accurate results for one or a specific set of modes. As a criteria forexpanding the VCI space, we can use a Fermi like resonance parameter

    = Hi, jHi,iH j, j

    (2.18)this means, that in our VCI implementation it is possible, while computing (screen-

    ing) the diagonal elementsHi,i, to include in our final reduced VCI space, only the con-figurations that are strongly coupled (according to Eq. 2.18) to the vibrational state(s)that we are interested in. This simple configuration selection approach is similar towhat has been very recently published by Rauhut [80]. Although, the configurationselection can be very effective in reducing the VCI space and yield accurate resultsfor fundamental states, as we will see in Section 2.6.1, the method has not yet beenthoroughly tested, especially for excited states.

    2.4.3 Numerical Integral Rules

    When writing a VMP2 or virtual VCI program, the evaluation of integrals between areference state n and a a virtual statem of the form

    (m)

    O(Q)(n), is an importantcomputational bottleneck. Therefore, special care has to be taken in implementing thiscalculation as efficiently as possible.

    As these computations are done in the framework of a hierarchical expansion (seeEq. (2.7)), using a product of one-dimensional vibrational wave functions as a basis set

  • 2.4 Correlation Correction 22

    ((n)) that we call here our reference state (see Eq. (2.1)), it is useful to remember afew rules for achieving optimal efficiency

    (1) modal vibrational wavefunctions are orthonormal, i.e. mii |nii = mi,ni(2) different vibrational modes interact only if there is a coupling potential that re-

    lates them. In other words, the integrals can be expanded in the following generalform

    (m)O(Q)(n)=

    mii

    i

    O(1)(Qi)nii

    N1

    l 6=i

    mll |nll

    +

    mii m jj

    i< j

    O(2)(Qi,Q j)nii n jj

    N2

    l 6=i, j

    mll |nll

    +

    mii m jj mkk

    i< j

  • 2.4 Correlation Correction 23

    From these rules, we can deduce that the VMP2 equation can be written in a numeri-cally efficient way as

    E(n)MP2 =pairs

    i, j

    mii m jj Vi jnii n jj 2nii +

    n jj

    mii

    m jj

    +triples

    i, j,k

    (2.23)

    Such techniques were implemented for some time in the thctk [81] program, but it isonly very recently that reports for such implementations in GAMESS-US [82], havebeen published [83, 84] for VMP2. In Python pseudo code, such an implementationfor the evaluation of Eq. (2.23) can be done efficiently as shown in Listing 2.1.

    Listing 2.1: Python pseudo code for VMP2

    1 eMP2 = 02 f o r i, j, Vi j i n TwoDimPoten t ia l :3 En = nii +

    n jj

    4 f o r mi i n [ 0 : max ]5 i f mi 6= ni :6 # p r e c o m p u t a t i o n o f Vnm7 f o r m j i n [ 0 : max ] :8 i f m j 6= n j :9 Em = mii +

    mij

    10 Vnm =

    mii m jjVi jnii n jj

    11 eMP2 += (VnmVnm) / ( EnEm )12 f o r i, j, k, Vi jk i n T h r e e D i m P o t e n t i a l :13 . . .

    Besides avoiding the evaluation of integrals which are zero as a result from our choiceof basis set, writing the VMP2 code as in Listing 2.1 has other advantages. In line2, we are not running over a fixed number of indices, but rather on a an object listTwoDimPotential containing the existing pair-potentials. Therefore, it is possible toeasily and efficiently add, remove or replace potentials. Further computational effi-ciency can be achieved by noticing that the integral Vnm can be precomputed (numericalcontraction) in the outer-loop on line 6.

  • 2.4 Correlation Correction 24

    Although the integral rules are in principle the same for VMP2 and VCI, writing anefficient code for the later is less straightforward. The main difficulty coming fromthe computation of the off-diagonal elements that are also the most time consumingpart. However, this problem can be tackled by recognizing from Eqs. (2.20), (2.21)and (2.22) the three possible types of off-diagonal VCI matrix elements that can occur.Using the dictionary data type built into Python, it is possible to avoid lengthy andcomplicated conditional loops that would be normally required.

    Listing 2.2: Python pseudo code for computing off-diagonal VCImatrix elements

    1 # b u i l d d i c t i o n a r i e s2 d e f O f f D i a g o n a l C I e l e m e n t ( (m),(n) )3 Hnm = 04 i f mini : # s i n g l y ( de ) e x c i t e d5 f o r j, Vi j i n S Tree 2D S [ i ] :6 Hnm +=

    mii m jj |Vi j|nii n jj

    7 f o r j, Sub Tree i n S Tree 3D [ i ] :8 f o r k, Vi jk i n ST :9 Hnm +=

    mii m jj mkk |Vi jk|nii

    n jj nkk

    10 e l i f mi,m jni,n j : # d o u b l y ( de ) e x c i t e d11 Hnm +=

    mii m jj |Vi j|nii n jj

    12 f o r j, Sub Tree 3D i n D Tree 3D [ i ] [ j ] :13 f o r k, Vi jk i n Sub Tree 3D :14 Hnm +=

    mii m jj mkk |Vi jk|nii

    n jj nkk

    15 e l i f mi,m j,mkni,n j,nk : # t r i p l y ( de ) e x c i t e d16 Hnm +=

    mii m jj mkk |Vi jk|nii

    n jj nkk

    17 r e t u r n Hnm

  • 2.5 Ab-Initio Potentials and VSCF 25

    2.5 Ab-Initio Potentials and VSCF

    2.5.1 Localization and Density Fitting Methods

    The accurate computation of the PESs is essential in the VSCF approximation, it isespecially true for the diagonal expansion V (1)i (Qi) which accounts for most of theanharmonicity. A major part of the computational effort at high ab-initio level hastherefore to be spent for its evaluation in order to achieve reasonable accuracy.

    Over the years, driven by the successful development of gradient corrected exchangeand correlations functionals, density functional theory (DFT) methods have been ac-cepted by the chemistry community as a cost-effective approach for the computing ofmolecular structures and vibrational frequencies. The main advantage of DFT is itsformal O(N3) scaling with the number of basis functions which makes it possible topredict energies and molecular structures of large molecules, compared to the O(N5)scaling of MP2 or even more dramatic O(N7) scaling of CCSD(T).However, DFT still suffers from certain deficiencies. First, there is no way to system-

    atically assess or to improve the accuracy of a calculation, since the exact functional isunknown. Thus, the results depend on the chosen functional, and experience is neededto select a suitable functional for a given problem. Second, most currently well testedDFT functionals do not account for dispersion, and for large systems it is anticipatedthat intramolecular dispersion type interactions will play a significant role [85, 86]. Fordescribing the conformational space of biomolecules, DFT is in general likely to be oflower accuracy than MP2 with an aug-cc-pVTZ basis set [87]. The latter is capable ofan accuracy of 1 kJ/mol, for relative conformational energies.

    On the other hand, post-HF methods like MP2 and CCSD(T) represent an estab-lished hierarchy to approach the exact solution of the electronic Schrodinger equation.Unfortunately, the delocalized character of the canonical Hartree-Fock orbitals, tradi-tionally used as a basis for correlation extensions, is responsible for the steep scalingwith respect to system and basis size. With the recent availability of methods likeDensity-Fitting (DF), also known as Resolution-of-the-Identity (RI) [88, 89], wherethe four-center two-electron integrals are replaced by linear combinations of three-center integrals through the introduction of an auxiliary basis expansion [90], and thelocalised orbital [91] MP2 approximations (LMP2) using local domains [92], an im-

  • 2.5 Ab-Initio Potentials and VSCF 26

    portant step has been made towards a significant reduction of the formal O(N5) scalingwith system size of the canonical MP2 energy.

    The combination of the local and DF methods at the MP2 level (DF-LMP2) with ana-lytical gradients [93] has been made available in the Molpro program [94] recently. Asshown by Hrenar and al. [95] the impact of DF-LMP2 on the harmonic vibrational fre-quencies is negligible. This is an important remark, as the DF-LMP2 approach allowsus to perform geometry optimization and normal mode analysis on large molecules at afraction of the computational cost in terms of hardware and time that would have beenotherwise required by the canonical approach. However, in the VSCF approach thegenerated PES requires non negligible displacements relative to the equilibrium geom-etry, which can be problematic for local correlation methods as the local orbital do-mains can change considerably with the geometry resulting in steps in the PES. Thisproblem and possible remedies have already been discussed in the literature [96, 97],but as the accuracy of the diagonal potential is crucial and localization methods canintroduce errors, we preferred here the DF-MP2 approach over the DF-LMP2.

    2.5.2 Multi-Level Calculations, Model Potentials andCoupling Norms

    It has been noted earlier that while the computation of the harmonic transition frequen-cies and low-order diagonal anharmonicities requires sophisticated ab-initio methodsto reach near spectroscopic accuracy, it is possible to obtain reasonably accurate inter-mode coupling potentials from simpler methods like semi-empirical or even chargeinteraction models [29]. This fact has been used widely in the simulation of multi-dimensional IR spectra where the experimentally more easily accessible transition fre-quencies have been used in combination with relatively simple models for the descrip-tion of the mode couplings.

    The hierarchical potential energy expansion in Eq. (2.7) allows for a straightforwardimplementation of such a multi-level scheme. In the VSCF algorithm any potential ofthe form V (n)i js(Qi,Q j, . . . ,Qs) is by construction zero at all points at which at least oneof the coordinates is zero. It is therefore possible to combine e.g. diagonal potentialsV (1)i (Qi) from higher level ab-initio calculations with coupling potentials V (2)i j (Qi,Q j)from a lower level computation. This multi-level scheme can of course also be applied

  • 2.5 Ab-Initio Potentials and VSCF 27

    to higher order coupling potentials [98]. In this work multi-level computations will beidentified by a short-hand notation of the form D:P:T where D describes the methodused to obtain the diagonal potential P the pair-coupling potentials and T the triplepotentials respectively.

    Due to the quickly increasing number of PES points required in the standard ap-proach, the applicability of VSCF to peptides and proteins is currently limited to PESexpansions based on classical force-fields. Even using a multiple level scheme, ac-curate ab-initio PES models are therefore inherently restricted to small molecules.Computational efficiency can be easily achieved by parallelizing i.e. distributing thepointwise PES evaluations over a pool of computers as done in this work. Neverthe-less, in order to be amenable for the simulation of vibrational spectra of larger peptidesand proteins, the generation of ab-initio PES must rely on a systematic procedure toneglect or approximate many coupling terms in the PES expansion of Eq. (2.7).In Chapter 4 we will see that if only certain specific vibrational states are of interest,

    then only a subset of couplings needs to be included at full accuracy. The magnitudeof the importance of these couplings is of course not known beforehand, therefore onewould need an efficient way to avoid unnecessary computations.

    In order to achieve this computational saving, we can use model potentials which giveus the possibility of mimicking the potential using only a few specific points on the V (2)i jor V (3)i jk potential. This allows us to either discard, use model potentials or evaluate thepotentials fully based on certain criterias. In the case of pair-coupling potentials, thefour points needed for modelling V (2)i j are chosen at displacements that corresponds tothe maximum of the first excited state of the respective diagonal potentials.

    The model function for an arbitrary point with coordinates (x1,x2, ) on the PES,where~r is a vector with ||~r||=

    x12 + x22 + , is of the form

    V (n)i j(~r) = F(~r) P()(r) (2.24)

    where F(~r) is an angular function that ensures that the potential goes to zero whenapproaching a coordinate axis

    F(~r) =(

    n

    ||~r||2

    )n ni=1

    x2i , F(0) = 0 (2.25)

  • 2.5 Ab-Initio Potentials and VSCF 28

    and P()(r) is a polynom where is defined by convention as

    (sign(x1),sign(x2), ), if x1 > 0(sign(x1),sign(x2), ), if x1 < 0 (2.26)

    this means that a polynomial function P() is equivalent to P(++) (see Figure 2.3). Inthis work we have chosen the polynomial function to be of fourth degree, where P(0) =0 by construction and we make the assumption that the first and second derivatives ofP(r) are zero. This means that the polynom is of the form P(r) = c||~r||3 + d||~r||4,where the coefficients c and d can be obtained by solving the problem for two points.For example a and b in Figure 2.3 will give us the coefficients of the polynom P(++)

    which will then be used to build the PES using Eq. (2.24) for any point situated in thetwo quadrants.

    Figure 2.3: Schematic representation of a four point model poten-tial

    With this very simple model one can already capture most of the properties of thePES with considerably fewer points. If working on a 8X8 grid this would mean that 64pointwise evaluations on the PES are needed, while using our model only 4 points areneeded to approximate the potential, achieving a non-negligible reduction of a factorof 16.

    The potentials in Figures 2.4 and 2.5 for the the pair-couplings of modes 1/3 of NMAand amide I/II respectively, are shown here as limiting cases. As expected, such amodel will have difficulties to represent potentials that would normally require the in-clusion of higher order terms in the polynomial approximation. Also, in the case of

  • 2.5 Ab-Initio Potentials and VSCF 29

    Figure 2.4: NMA PES plot for pair-coupling potential of amideI/II, model (left) and full PES (right)

    Figure 2.5: NMA PES plot for pair-coupling potential of modes1/3, model (left) and full (right)

  • 2.5 Ab-Initio Potentials and VSCF 30

    vibrational modes with bending or torsional character couplings are typically overesti-mated as the model potential does not decay quickly enough to zero when approachingthe coordinate axis compared to the true potential.

    Without any a-priori knowledge of the coupling potentials and their importance forthe spectrum, any approach that depends on the selection of coupling potentials re-quires an appropriate definition of a coupling strength or a coupling potential normsimilar to the simple Frobenius norm. A reasonable definition of such a norm shouldaccount for the accessibility of the coupling potential, i.e. include the wave function ofthe system as a weighting factor.

    This leads naturally to an integral of the form (k)i (l)j |V

    (2)i j |

    (k)i

    (l)j or higher order

    equivalent. These integrals contribute directly to the total energy of the vibrationalsystem if the (k)i are the converged VSCF single particle functions [98]. A relatedquantity has been used before to estimate coupling strengths that mediate intramolec-ular energy redistribution (IVR) processes [99].The evaluation of the above mentioned integrals presupposes the knowledge of the

    full VSCF wave function which is not available before the PES expansion has beendefined. We will use instead the eigenfunctions of the corresponding one-dimensionalsub-problems

    (Ti +V

    (1)i

    )(k)i =

    (n)i

    (k)i (2.27)

    in order to define the norms in the following

    V (2)i j kl =(k)i (l)j |V (2)i j |(k)i (l)j (2.28)

    The computation of the V (2)i j kl norm according to Eq. (2.28) requires the values ofthe coupling potential on a full (Qi,Q j) grid for the evaluation of the integral. As this isthe most computationally demanding operation, it is necessary to approximate the inte-grals by simpler expressions. The norms V (2)i j 11 can be reasonably approximated as

    V (2)i j 11 Ni j = |(biai)(b j a j)| (x,y){ai,bi}{a j,b j}

    (1)i (x)(1)j (y)V

    (2)i j (x,y) (2.29)

  • 2.6 VSCF Implementation and Benchmarking 31

    here, ai and bi are the positions of the extrema of the corresponding one-dimensionalwave function (1)i . In this approximation only four evaluations of the coupling poten-tial are needed to compute Ni j. These four points are also used to build, if required,our model potential.

    2.6 VSCF Implementation and Benchmarking

    A VSCF algorithm with MP2 and CI correlation correction has been implemented ina package for Theoretical Chemistry [81] written in Python [100] an interpreted, inter-active, object-oriented programming language. Although a VSCF algorithm is freelyavailable [101] through the GAMESS-US [82] program, using a program written ina high-level language such as Python offers many advantages. Perhaps the most im-portant ones with respect to efficiency are the possibility to do coarse-grain parallelprocessing of the PES grid points by interfacing to any external program that can eval-uate such points as well as a efficient and flexible CI implementation. Thanks to theversatility offered by a high-level scripting language, tailoring very specific CI Hamil-tonians can be done in a trivial and efficient way.

    2.6.1 Formaldehyde

    In order to asses our implementation, we compare our results for formaldehyde [H2CO]with the results obtained by Romanowski et al. [74] and those of Christiansen [71] fromyet another VSCF implementation. The formaldehyde potential is based on a quarticforce field given in Ref. [74]. Our VSCF results are based on a grid representationusing a modal basis obtained from the solutions of the effective one-dimensional vi-brational problems on a collocation grid [60, 62, 63, 65].We used an equally spaced, symmetrical 16 point grid of width 40 (dimensionless

    coordinate units) for formaldehyde. With these values the VSCF energies were con-verged within less than 1 cm1 for all states shown in Table 2.1. For the VMP2 andVMP2-m perturbation approaches we used a maximum excitation level of 5. Threedifferent VSCF-VCI computations with no restriction on the type of excitations (i.e.up to hexuple-excitations) are shown in Table 2.1 namely

  • 2.6 VSCF Implementation and Benchmarking 32

    (a) A small CI calculation using a maximum excitation level of ni = 5 for everymode with the restriction for the sum of excitations in a configuration of Nmax =

    6i=1

    ni = 8, resulting in a matrix of dimension 2835. The state energies were

    obtained by diagonalisation of a CI matrix built using the ground state VSCFmodal wavefunctions as a basis set.

    (b) In this case, we used a maximum excitation level of ni = 6 with Nmax = 12resulting in a matrix of dimension 15792. The state energies in this case wereobtained by diagonalising for every state a CI matrix built from its correspondingVSCF modal wavefunctions (state specific CI).

    (c) The choice for the reference calculation was ni = 6 and and no restriction onNmax resulting in a large CI matrix of dimension 117649. These values are suf-ficiently large to consider our CI energies converged. The state energies wereobtained by diagonalisation of a CI matrix built using the ground state VSCFmodal wavefunctions as a basis set.

    On an AMD 64 bit Opteron 240, it took approximatively 1 minute, 40 minutes and5 hours to build a CI matrix for (a), (b) and (c) respectively. In Figure 2.6 the pair-coupling potentials and their corresponding norms

    V (2)i j kl are plotted as defined inEq. (2.28). This representation helps us to visually identify the coupling pattern be-tween vibrational modes. Here, vibrational modes are numbered from lowest to highestfrequency, which corresponds to CH2 wagging, CH2 asymmetric bend, CH2 symmet-ric bend, C=O stretch, CH2 symmetric stretch and CH2 asymmetric stretch. As we cansee from Figure 2.6 the strongest couplings are the CH2 symmetric bending and C=Ostretch with CH2 the symmetric stretch, while the wagging mode is coupled to all othervibrational modes.

  • 2.6 VSCF Implementation and Benchmarking 33

    Table 2.1: VSCF energy levels of Formaldehyde (cm1). The CIcorrelation energies for the vibrational states were ob-tained solving the CI matrix of (a) the ground statewavefunction with Nmax = 8 and ni = 5, (b) state specificwith Nmax = 12 and ni = 6 (c) the ground state wavefunc-tion with ni = 6 and no restriction on Nmax.

    State Nr State EV SCF EVMP2corr EVMP2-mcorr EVCI (a)corr EVCI (b)corr EVCI (c)corr

    1 0,0,0,0,0,0 5797.4286 -18.4151 -18.3964 -19.0382 -19.04337 -19.09372 1,0,0,0,0,0 8612.6714 -55.0177 -54.7800 -54.1810 -54.43382 -54.48433 0,1,0,0,0,0 7550.2275 -23.0412 -23.0208 -25.6888 -25.26937 -25.77134 0,0,1,0,0,0 7305.5905 -26.2882 -26.2644 -27.7348 -27.25432 -27.81005 0,0,0,1,0,0 6948.8424 -23.1534 -23.1287 -24.5906 -23.07735 -24.48266 0,0,0,0,1,0 8644.3184 -8.0563 -22.5043 -31.3919 -44.01901 -31.72827 0,0,0,0,0,1 7046.8657 -25.7464 -25.7230 -26.7580 -25.76608 -26.79988 2,0,0,0,0,0 11374.2897 -108.7687 -106.4064 -103.9836 -108.33861 -105.35069 0,2,0,0,0,0 9288.3211 -26.1917 -26.1738 -33.6774 -32.60271 -34.0981

    10 0,0,2,0,0,0 8811.1101 -34.2611 -34.2264 -36.9792 -36.31016 -37.377511 0,0,0,2,0,0 8108.0572 -43.0828 -42.6676 -43.1819 -36.72622 -41.123912 0,0,0,0,2,0 11422.9704 -21.4780 -37.0378 -18.9403 -55.66591 -22.807513 0,0,0,0,0,2 8301.5215 -43.7098 -43.3130 -44.0155 -39.98756 -43.632714 1,1,0,0,0,0 10363.6020 -60.8190 -60.5749 -61.5551 -61.42239 -62.018015 1,0,1,0,0,0 10087.4365 -52.6557 -53.7557 -66.3862 -63.31272 -66.906316 0,1,1,0,0,0 9057.6948 -35.2749 -35.2409 -39.7986 -38.62077 -39.936217 1,0,0,1,0,0 9718.5912 -43.0302 -44.4150 -42.7424 -42.89459 -43.365118 0,1,0,1,0,0 8693.7164 -28.9101 -28.8815 -33.6726 -31.52913 -33.474119 0,0,1,1,0,0 8451.4832 -32.1478 -32.1163 -35.0536 -32.36494 -34.945820 1,0,0,0,1,0 11465.5496 -165.5533 -161.6654 -170.9214 -175.03016 -173.378521 0,1,0,0,1,0 10394.1078 -34.3486 -44.5943 -50.5386 -59.88074 -51.017022 0,0,1,0,1,0 10106.4052 240.4892 26.3043 -10.6940 -31.67074 -11.447723 0,0,0,1,1,0 9735.6612 208.4242 -3.1567 -14.1731 -34.50069 -14.456424 1,0,0,0,0,1 9829.5772 -58.7981 -59.8331 -54.6330 -54.75017 -55.351725 0,1,0,0,0,1 8795.3164 -1.6357 -8.5243 -27.1289 -26.04724 -27.280526 0,0,1,0,0,1 8562.6086 -90.6171 -76.8317 -69.7947 -65.97593 -69.871427 0,0,0,1,0,1 8202.8989 -31.1141 -31.0835 -32.6872 -30.09752 -32.726128 0,0,0,0,1,1 9855.0704 256.5636 -5.1797 -14.8059 -34.09379 -14.9452

    rmsd, with respect to (c) 54.11 82.20 10.10 1.03 9. 61

  • 2.6 VSCF Implementation and Benchmarking 34

    From the results shown in Table 2.1 and the root mean square deviation (RMSD) itis obvious that VMP2-m represents a substantial improvement over VMP2 in terms ofstability and accuracy especially for the different states involving modes 1 and 5 whichare problematic, as pointed out in the previous studies [71, 74], due to the importantinteraction with several other states. Comparing with (c) results (see Table 2.1(c)), wecan see that the errors introduced by the canonical VMP2 can sometimes shift thevalues by more than 100 cm1 while the VMP2-m corrections are on average closer tothe VSCF-CI results.

    1 2 3 4 5 6

    1

    2

    3

    4

    5

    6

    Figure 2.6: Formaldehyde pair-couplings V (2)i j and correspondingnorms mapping. Upper left triangle: Pair-potentialterms. The potential energy values have been clippedat 0.02Eh and are color coded with a rainbow paletteranging from violet (0.02Eh) to red (+0.02Eh).Lower right triangle: Relative norms of the couplingpotentials

    V (2)i j 00 as defined in Eq. (2.28). The max-imum norm has been scaled to 1 and the color codingon the interval [0,1] ranges from green to red.

    Our large VSCF-CI computation (see Table 2.1(c)) reproduces the results of both Ro-manowski [74] and Christiansen [71]. The deviation of states number 5 and 6 fromBowmans results, which is also observed in the results of Christiansen, can be at-tributed to unconverged energies due to a more restrictive CI space in Bowmans study.The small CI calculation of Table 2.1(a), is already sufficient to converge all the statesincluding the problematic doubly excited ones that involve modes 1 and 5, to an accu-racy of approximatively one wavenumber compared to the large VSCF-CI calculation.Considering that the various errors introduced during the evaluation of the PES are

  • 2.6 VSCF Implementation and Benchmarking 35

    larger, this order of accuracy can be considered sufficient.

    The state specific VSCF-CI computations in Table 2.1(b) do not display a faster con-vergence to the limit values of the large VCI computation as one might expect. In fact,we increased the size of the CI space in order to improve correlation corrections. Asalready discussed in Ref. [71], near degeneracies that causes several Hartree productsto have a major contribution in the wave function can make state-specific methods witha pure mode-excitation criteria for the definition of the excitation space, to have slowerinitial convergence of the energy with respect to the excitation level.

    In Table 2.2 we show our results for the configuration selection VCI method. Beforediscussing the results, we would like first to make some comments on the table. Twodifferent types of configuration selective VCI are shown here, a ground state (GS) astate selective (SS), see Section 2.4.2. Starting from a search space that includes onlysingle and double excitation up to the 6th excited state i.e. (6,6), the configurationsearch space is, in every (double) column, increased up to hexuple excitations to the6th excited state. The threshold parameter (see Eq. 2.18) was set to 107. Withthis value the VCI results are considered to be converged, since for lower values thechanges are insignificant.

    Comparing our results obtained using a large VCI matrix (Table 2.1) with the onesobtained using a configuration selection approach (Table. 2.2), we can see that the GSresults for the fundamentals are essentially converged with a search space that includesquadruple excitations up to the 6th excited state. For every fundamental vibrationalstate, only about 500 states have to be included to make the correlation energy con-verged to within less than 2 cm1. The SS results show also the same pattern of con-vergence with typically only a few more states required in the VCI expansion spacethan previously. However, as already discussed above, problematic cases like statenr. 6 are not converged.

    For doubly excited states and combination bands, our configuration selection schemeseems to be somewhat less reliable, with larger and unevenly distributed deviationscompared to the converged VCI results of Table 2.1. They require at least quintupleexcitations in order to be considered converged and the vibrational states require about600 terms in their configuration space, which is slightly more than for the fundamen-tals. The restricted VCI space seems also to be unable to handle reliably the mostproblematic states the like the overtone vibrational state nr. 12.

  • 2.6 VSCF Implementation and Benchmarking 36Ta

    ble

    2.2:

    Form

    alde

    hyde

    VCI

    corr

    elat

    ion

    ener

    gies

    usin

    ga

    config

    -

    ura

    tion

    sele

    ctio

    nsc

    hem

    e.

    GS:

    Gro

    und

    stat

    e,SS

    :st

    ate

    spec

    ific,

    (i,j,.

    ..):

    config

    u-

    ratio

    ns

    sear

    chsp

    ace

    Sear

    chSp

    ace

    (6,6)

    (6,6,

    6)(6,

    6,6,

    6)(6,

    6,6,

    6,6)

    (6,6,

    6,6,

    6,6)

    Stat

    eN

    rSt

    ate

    GS

    SSG

    SSS

    GS

    SSG

    SSS

    GS

    SS1

    0,0,

    0,0,

    0,0

    -17

    .52

    9-17

    .52

    9-18

    .57

    8-18

    .57

    8-18

    .89

    6-18

    .89

    6-18

    .89

    6-18

    .89

    6-18

    .89

    6-18

    .89

    62

    1,0,

    0,0,

    0,0

    -49

    .02

    5-49

    .63

    9-52

    .37

    5-52

    .77

    8-53

    .37

    3-53

    .75

    9-53

    .38

    4-53

    .77

    0-53

    .38

    4-53

    .77

    03

    0,1,

    0,0,

    0,0

    -7.

    290

    -7.

    0424

    -24

    .56

    6-24

    .15

    3-25

    .32

    2-24

    .89

    6-25

    .57

    2-25

    .13

    7-25

    .57

    2-25

    .13

    74

    0,0,

    1,0,

    0,0

    -8.

    171

    -8.

    3296

    -26

    .41

    3-25

    .93

    1-27

    .45

    9-26

    .94

    2-27

    .56

    1-27

    .04

    6-27

    .56

    1-27

    .04

    65

    0,0,

    0,1,

    0,0

    -6.

    486

    -5.

    9080

    -22

    .98

    9-21

    .51

    0-24

    .13

    8-22

    .66

    1-24

    .51

    8-22

    .98

    9-24

    .51

    8-22

    .98

    96

    0,0,

    0,0,

    1,0

    -18

    .23

    4-70

    .24

    02-23

    .83

    4-15

    1.31

    0-30

    .30

    5-16

    6.07

    -30

    .30

    5-16

    6.07

    -30

    .30

    5-16

    6.07

    47

    0,0,

    0,0,

    0,1

    -6.

    735

    -6.

    5606

    -25

    .72

    1-24

    .81

    1-26

    .68

    5-25

    .68

    0-26

    .68

    5-25

    .68

    0-26

    .68

    5-25

    .68

    08

    2,0,

    0,0,

    0,0

    -19

    .11

    4-94

    .18

    20-92

    .96

    9-13

    8.82

    0-96

    .24

    3-18

    0.88

    9-96

    .25

    7-18

    1.04

    0-96

    .25

    7-18

    1.04

    09

    0,2,

    0,0,

    0,0

    -11

    .60

    1-10

    .96

    06-32

    .19

    5-31

    .09

    6-33

    .31

    3-32

    .15

    1-33

    .58

    1-32

    .39

    8-33

    .58

    1-32

    .39

    810

    0,0,

    2,0,

    0,0

    -11

    .27

    0-11

    .89

    18-34

    .24

    7-33

    .49

    8-36

    .57

    0-35

    .67

    9-36

    .71

    0-35

    .82

    7-36

    .71

    0-35

    .82

    711

    0,0,

    0,2,

    0,0

    -23

    .89

    5-19

    .00

    39-41

    .18

    7-34

    .68

    0-42

    .47

    9-35

    .94

    7-42

    .93

    7-36

    .27

    3-42

    .93

    7-36

    .27

    312

    0,0,

    0,0,

    2,0

    98.47

    913

    1.82

    5723

    2.42

    6-11

    7.55

    412

    4.45

    8-15

    4.75

    112

    4.45

    8-15

    4.75

    112

    4.45

    8-15

    4.75

    113

    0,0,

    0,0,

    0,2

    -16

    .38

    9-14

    .17

    88-40

    .59

    8-37

    .11

    7-43

    .44

    5-39

    .33

    6-43

    .44

    5-39

    .33

    6-43

    .44

    5-39

    .33

    614

    1,1,

    0,0,

    0,0

    -24

    .52

    8-26

    .98

    47-56

    .35

    2-56

    .50

    9-59

    .19

    4-59

    .08

    7-59

    .82

    5-59

    .71

    0-59

    .82

    5-59

    .71

    015

    1,0,

    1,0,

    0,0

    -22

    .90

    1-23

    .32

    93-60

    .35

    8-57

    .93

    5-63

    .63

    5-60

    .88

    5-64

    .09

    5-61

    .32

    7-64

    .09

    5-61

    .32

    716

    0,1,

    1,0,

    0,0

    -13

    .57

    5-14

    .41

    09-19

    .52

    3-19

    .06

    6-38

    .67

    4-37

    .33

    8-39

    .44

    7-38

    .06

    3-39

    .44

    7-38

    .06

    317

    1,0,

    0,1,

    0,0

    -3.

    238

    -8.

    8606

    -36

    .51

    8-37

    .79

    1-39

    .09

    3-40

    .14

    4-40

    .10

    8-41

    .11

    5-40

    .12

    1-41

    .12

    618

    0,1,

    0,1,

    0,0

    0.40

    6-6.

    9918

    -15

    .21

    8-13

    .97

    8-32

    .43

    3-30

    .24

    9-33

    .27

    5-31

    .06

    9-33

    .57

    2-31

    .31

    819

    0,0,

    1,1,

    0,0

    3.86

    3-5.

    4141

    -15

    .18

    9-14

    .04

    9-33

    .57

    0-30

    .91

    0-34

    .76

    4-31

    .96

    7-34

    .89

    5-32

    .07

    220

    1,0,

    0,0,

    1,0

    -94

    .88

    9-16

    1.43

    015

    5.28

    1-20

    4.25

    9-16

    7.61

    1-23

    2.07

    4-16

    7.71

    2-23

    2.50

    9-16

    7.71

    2-23

    2.50

    921

    0,1,

    0,0,

    1,0

    35.99

    9-17

    .15

    6-38

    .52

    4-86

    .50

    7-43

    .35

    8-15

    4.95

    5-48

    .96

    9-16

    7.49

    9-48

    .97

    1-16

    7.50

    122

    0,0,

    1,0,

    1,0

    111.

    102

    -38

    .54

    04.

    914

    -14

    4.35

    8-7.

    476

    -16

    3.80

    0-7.

    735

    -16

    3.97

    3-7.

    735

    -16

    3.97

    323

    0,0,

    0,1,

    1,0

    51.98

    7-5.

    861

    17.00

    1-70

    .30

    01.

    102

    -12

    5.96

    9-2.

    936

    -13

    6.92

    5-10

    .42

    6-13

    6.92

    824

    1,0,

    0,0,

    0,1

    -11

    .96

    4-28

    .25

    0-47

    .97

    7-48

    .92

    7-52

    .30

    4-52

    .80

    8-52

    .37

    0-52

    .86

    8-52

    .37

    0-52

    .86

    825

    0,1,

    0,0,

    0,1

    7.17

    0-1.

    083

    -8.

    379

    -8.

    288

    -26

    .07

    1-25

    .11

    6-26

    .74

    2-25

    .65

    1-26

    .74

    2-25

    .65

    126

    0,0,

    1,0,

    0,1

    -39

    .25

    0-43

    .79

    7-55

    .77

    6-52

    .08

    9-68

    .61

    4-64

    .78

    1-68

    .92

    3-65

    .04

    1-68

    .92

    3-65

    .04

    127

    0,0,

    0,1,

    0,1

    11.05

    7-2.

    308

    -12

    .67

    2-11

    .50

    5-31

    .57

    5-29

    .12

    2-32

    .70

    2-29

    .99

    4-32

    .70

    2-29

    .99

    428

    0,0,

    0,0,

    1,1

    75.80

    9-67

    .75

    5-0.

    006

    -17

    5.92

    3-13

    .34

    2-19

    1.15

    3-13

    .38

    9-19

    2.34

    2-13

    .38

    9-19

    2.34

    2

  • 2.6 VSCF Implementation and Benchmarking 37

    Although we have presented a very simple selection scheme for constructing a VCIconfiguration space for formaldehyde, the results are very encouraging. We have con-firmed the importance of quadruples in the VCI expansion for obtaining reliable fun-damental states, as also observed in Ref. [80] but also the importance of quintuplesfor higher excited states. This advocates the use of configuration selection schemes inVCI studies. However, the VCI development is still at an early stage and more studiesand experimental results are required in order to analyze and develop more efficientselection schemes for improving the reliability of the method.

  • 38

    Chapter 3

    Formamide and Thioformamide(The Importance of Triples V (3)i jk )

    3.1 Introduction

    As the simplest molecule containing the characteristic HNCO (peptide) linkage, theimportance of formamide [HCONH2] as a model compound is obvious. Its shallowPES for the out of plane wagging mode that allows the amide hydrogens to moveout of the plane has been the source of recurrent debate about its structure (planaror not) for many years (see for instance Ref. [102] and references therein). Basedon a relatively large rotational barrier around the CN bond (estimated to be about 15-20 kcal/mol), several studies seem to have achieved a consensus about its double bondcharacter [103, 104]. This is even more pronounced for the sulfur analogue, thiofor-mamide [HCSNH2], and can be satisfactorily explained using a resonance model [105].The symmetric and asymmetric NH stretches are heavily coupled to low frequency

    modes implying that vibrational excitations in the NH stretching region consist ofrather complicated superpositions of several normal modes. Any excitation of theNH stretch will therefore effectively deposit energy in the low frequency modes ofthe molecule [102]. These strongly coupled low frequency modes are for instance sus-pected to be involved in the ultrafast relaxation of the amide-I mode [106]. This wealthof material on formamide is in contrast to the situation for its sulfur analogue. To ourknowledge thioformamide has been studied in detail only recently by Kowal [107] atthe VSCF level.

  • 3.2 Computational Methods 39

    In several recent studies of small and medium size molecules [99, 107, 108] triplesV (3)i jk (see Eq. (2.7)) are often neglected (pairwise approximation) but so far no detailedstudy has been done to study the consequences. The relatively small size of these twomolecules (12 fundamental modes) allows us to use a hierarchical expansion up totriple potentials V (3)i jk at an accurate ab-initio level (MP2 level with a triple zeta qualitybasis set) and study their importance and impact on the VSCF results.

    3.2 Computational Methods

    For formamide and thioformamide, geometry optimisation and normal mode analy-sis where carried out using the Molpro [94] program at the MP2/aug-cc-pVTZ levelwithout symmetry constraints, resulting in planar structures that are true minima. Theanharmonic frequency computation using vibrational second order perturbation the-ory (see Section 1.2.2) was carried out using the Gaussian 03 [41] program at theMP2/aug-cc-pVTZ level with tightly optimized geometries. The differences in the op-timized geometries and harmonic normal mode analysis with Molpro and Gaussian aresufficiently small to be considered equivalent.

    The PES point evaluations necessary for a VSCF computation were obtained us-ing Molpro at the DF-MP2/aug-cc-pVTZ and CCSD(T)/aug-cc-pVTZ level for thediagonals V (1)i , at the DF-MP2/aug-cc-pVTZ for the pairs V

    (2)i j , and finally at the DF-

    MP2/cc-pVTZ level for triples V (3)i jk . The pointwise PES evaluations were based on a8 point direct product grid. The PES is then interpolated to a denser 16 point grid us-ing splines. Vibrational-rotational contributions were not included in the anharmonicVSCF treatment.

    The VCI matrix of formamide included single, double and triple excitations up to the6th excited state, quadruple excitations up to the 3rd and quintuple excitations up to thesecond excited state, for a total of 369145 configurations. For the thioformamide VCImatrix we used the same procedure but without quintuple excitations yielding a totalof 176689 configurations. Experimental frequencies for formamide were taken fromRef. [109]. Unfortunately, to this day there are still only partial experimental valuesavailable for thioformamide in the gas phase [110].

  • 3.3 Results and Discussion 40

    3.3 Results and Discussion

    3.3.1 Formamide

    Before discussing the influence of different approximations and coordinate systems onthe VSCF method and its correlation extensions, we would first like to establish whatlevel of accuracy can be ideally expected for a VCI treatment and how this compares tothe results from the PT2 method. In the following we will compare our computationsto the experimental fundamental vibrational frequencies for formamide taken fromRef. [109]. The results of a PT2 computation based on MP2/aug-cc-pVTZ energies areshown in Table 3.2. These have to be compared to the corresponding VCI results in thesecond to last column of Table 3.3. We have to note as a first result that PT2 completelyfails to reproduce the low frequency NH2 wagging mode. It exhibits a very large un-realistic anharmonic blueshift (> 1000 cm1). This is a problem for the out of planemotion for amide groups that has already been pointed out by Barone [40]. We willtherefore discard this outlier mode from the PT2 results in the following discussion.The PT2 root mean square deviation (RMSD) w.r.t. the experimental frequencies ofthe remaining modes (21.2 cm1) seems to be comparable to the corresponding VCIresult in Table 3.4 (23.1 cm1). This is mainly due to the excellent agreement of thePT2 results with the experimental frequencies for the NH stretch modes, but also withmodes exhibiting a negligible degree of anharmonicity (e.g. NCO bend, mode 2). Forthese particular vibrations we can take the PT2 frequencies as a complementary resultto the VCI computation since these modes are harder to converge in VCI. On the otherhand, the maximum absolute error of PT2 of 62 cm1 for the CH stretch (mode 10)is significantly larger than for VCI (47 cm1, also for CH stretch). The errors in VCIin general seem somewhat more evenly distributed and systematic than in PT2 (seeFigure 3.1).The good performance of PT2 for some modes is not indicative though of the quality

    of the semi-quartic force-field representation of the PES. It is rather a fortuitous cancel-lation of errors in the non-variational perturbation method. This was already reportedearlier by Carter et al. [111, 112] and shows that quartic force fields do not generallydescribe the anharmonicity of the potential with sufficient accuracy. Moreover, in thecase of strong resonances like for the NH2 wagging mode, simple perturbation the-

  • 3.3 Results and Discussion 41

    ory can give unrealistic deviations. In the last three columns of Tab. 3.2, where VSCFand VMP2 results based on the semi-quartic PT2 force-field are presented, the RMSDvalue of the variational method is larger (39.6 cm1) than for PT2. Adding correlationeffects shows that the VMP2-m approach of eq. 2.13 yields a lower RMSD (19.2 cm1)and less erratic deviations from the exact frequencies than plain VMP2 (52.0 cm1).The variational procedure yields more accurate results for the CH stretch (mode 10)and CH ipb (mode 7). The VMP2-m is method, is especially capable of producingaccurate results for strongly coupled modes like the NH2 wagging and the CH stretch.

    Table 3.1: Definitions of vibrational acronyms

    s stretch sb symmetric bendas asymmetric stretch sci scissoringb bend ip in-planess symmetric stretch opb out-of-plane bendab asymmetric bend tors torsion

    Table 3.2: Vibrational frequencies of Formamide in cm1, basedon the quartic force field MP2/aVTZ (for Symbols, seeTable 3.1). The outlier mode 1 of PT2 is not included

    Assignment Mode Nr. Harmonic Int.(km/mol) PT2 VSCF VMP2 V