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Ph D. Thesis on anharmonic corrections to vibrational modes
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
Summary
The rapid developments in laser technology and their use in multidimensional vibrational 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 subpicosecond 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 intermolecular interactions has made IR vibrational spectroscopy an established tool in thestudy of molecular matter. In particular for peptides and proteins, where the IR absorption in the spectral range of 14001800 cm1 (amideI 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 inhomogeneously broadened vibrational transitions in a narrow spectral region.
The unprecedented potential of multidimensional vibrational IR spectroscopy to disentangle 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 manybodyexpansion of the potential energy surface (PES) can provide us with an efficient framework 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 achieving 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 AbInitio Potentials and VSCF . . . . . . . . . . . . . . . . . . . . . 25
CONTENTS ii
2.5.1 Localization and Density Fitting Methods . . . . . . . . . . . 25
2.5.2 MultiLevel 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 Nmethylacetamide as a Model for Peptide Linkage 58
4.1 NMethylacetamide . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1.2 Computational Methods . . . . . . . . . . . . . . . . . . . . 60
4.1.3 Vibrational Spectrum . . . . . . . . . . . . . . . . . . . . . . 61
4.1.4 MultiLevel 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 FirstPrinciple Study of Hydrogen Bonded Complexes 90
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2 Hybrid DFT Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3 MultiCoefficient 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 threedimensional structure of proteins in general and the foldingevents in particular is of prime importance for understanding their biological function. 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 correctthreedimensional 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 arrangement reached on the time scale of seconds [2]. Nonbonded interactions, mainly hydrogen bonds, play a significant role and in many cases the equilibrium fluctuations aroundthe timeaveraged structure are essential to the proteins functionality [3]. Therefore,a short experimental timescale is essential in order to probe structures which dependon interactions that constantly rearrange, dissociate and reform under ambient conditions [4].Although in the past years, traditional Xray diffraction and nuclear magnetic res
onance (NMR) spectroscopy provided (and still provides) a wealth of structural information [5], they have known limitations. While the former can only give a staticpicture corresponding to a solidstate structure1 (therefore, certain classes of proteins
1 This is partially true, smallangle Xray scattering (SAXS) techniques can be used for the structural determination of proteins in solutions [6] but then only with a lowresolution (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 growing possibilities to control optical and IR pulses on those time scales have allowedthe recent successes in the experimental realization of coherent multidimensional IRspectroscopy. Particularly, it opened the exciting possibility to study the molecularstructure and dynamics of polyatomic systems in solution on timescales 6 to 10 orders of magnitude faster than NMR spectroscopy. With a temporal resolution down tothe subpicosecond regime [8, 9, 10], multidimensional 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 midinfrared region (800 3400cm1) of a tenresidue 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 amideI vibrationalmode (found at 1500 1800 cm1) which involves mainly the C=O stretch coordinatewith contributions from the CN stretch and NH bend motions (see Figure 1.2) is, atleast from an experimental point of view, the most important one due to its large transition dipole moment and sensitivity to secondary structure. Other important bandsare the amideA/B (3000 3500 cm1, essentially NH stretch) and amideII (1500 1550 cm1, mainly inplane NH bending) .Conventional onedimensional 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 amideIband lineshapes that corresponds to the different secondary structural elements encountered in proteins [16, 17], is in general only sufficient to qualitatively or semiquantitatively assign a relative ratio of secondary structure motifs (helix vs. sheet)contained in a particular protein2.
Figure 1.2: AmideI, 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 effects such as conformational fluctuations and their coupling to the local environment.In nonlinear 2DIR spectroscopy the structural and dynamical information is typicallypresent in terms of diagonal and crosspeak 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 amideI and amideII absorptions in the spectral 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 spectroscopy [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 amideI oscillator that couples electrostatically 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 relevant vibrations. This is a very restrictive approach, while in reality the mechanismsof vibrational energy transport in peptides are still not well understood. More complex 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 throughspace or
throughbond [21, 23, 24, 25, 26, 27]. Throughspace interactions are typically electrostatic in nature and have been successfully described using multipole or even dipoledipole interaction models [15, 28, 29]. In comparison, the throughbond interactionsare typically short ranged and need to be described employing abinitio quantum chemical (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 widespread use of vibrationalmodel Hamiltonians based on the assumption that the amideI 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 CH
vibration has been suggested as a secondary structure sensitive probe [31], showingthat a detailed study of CH vibrations in oligopeptides is also necessary along with themore commonly studied modes. However, in the case of a CH 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 vibrational 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 nonrotating (J = 0) quantummechanical vibrationrotation Hamiltonian3 (Watson Hamiltonian) [32, 33] in atomic units (a.u.) with massscaled normalmode coordinates in the BornOppenheimer (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 vibrationrotation Coriolis interaction (Tc) as well as the massdependent terms (Tw, also called Watson term) to the vibrational Hamiltonian are certainly negligible in our conclusions as both contributions scale with the inverse of themoments of inertia tensor of the whole molecule. The approximations in the treatment 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 rovibrational 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 assume 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 subsequently transformed into massweighted 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 routinely by most quantum chemistry packages. Nevertheless, the vibrational harmonicoscillator is a highly idealized system, where the eigenstate energy of a single vibrational 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 amplitude, irrespective of the amount of energy injected into the system and only the transitions that satisfy the vibrational selection rule n1 are allowed5. Consequently, overtones 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 quantummechanical anharmonic oscillator using a second orderperturbative treatment and the potential expansion of Eq. (1.4) up to quartic force constants has been known for some time now [34, 35, 37, 38]. Nevertheless, it is onlyrecently that anharmonic vibrational frequency computations using second order perturbation 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 nonzero.
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 ith harmonic frequency. The anharmonic constants xi j can be obtained 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 perturbativevariational approach 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 semirigid molecules,it should be used with care for highly anharmonic and floppy molecules like peptidesand proteins.
In a threepulse photonecho 2DIR spectrum as shown in the schematic in Figure 1.3 (left) for two coupled oscillators i and j, there are five peaks for each oscillator. The diagonal peaks labelled (a) and (b) are separated by the diagonal anharmonicity i, while the crosspeaks labelled (c) and (d), are separated by the offdiagonal
1.2 The Vibrational Problem 9
Figure 1.3: (Left) 2DIR 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 2DIR spectrum provides us with adirect measure of the anharmonicity. For proteins, peptides, and peptide model systemsthe amideI modes diagonal anharmonicities i typically range from 10 to 20 cm1 andup to ca. 160 cm1 for amideA vibrational modes. The mixedmode anharmonicityi j of neighboring amideI modes of an helix is approximatively 9 cm1 [43]. Allthe difficulty now, is in obtaining these anharmonic constants from first principle calculations. Assuming that in Eq. (1.9) all terms higher than bilinear interaction are negligible, 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 abinitio 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 amideI, interacts either throughspace or through bond with neighboring amide units. Separating the amideI vibrational manifold of states from all other vibrational degrees of freedom, this excitationcan move from site to site, thus giving rise to a delocalized excitonstate i.e. a vibrational 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 commutation relation
[Bi,B
j]= i j. The harmonic intermode coupling is designated by Ji j.
In other words, an amideI 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 abinitio computations. Inthe simplest models, the amideI 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 amideI 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 highresolution 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 vibrational 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 application. 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 reallife systems, heavily depend 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 successfully describe the vibrational multidimensional spectra of biomolecules, i.e. yield accurate vibrational force constants beyond the harmonic approximation for a wide rangeof bond types with a correct description of nonbonded interactions, perform well closeto the equilibrium structure as well as in nonequilibrium 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 multidimensional IR spectra by sumoverstates or propagation 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 timedependent multidimensional spectra. Recently, FranckCondon 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 onedimensional 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 massweighted 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 singlemode 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 ith coordinate. The resulting set ofVSCF equations has to be solved iteratively until selfconsistency 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 approximation 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) energies, respectively first and second part in Eq. (2.6).Choosing an appropriate onedimensional basis set (i) for each vibrational motion is
primordial, as they define the building blocks of the anharmonic vibrational wave function 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 onedimensional Schrodinger equations through the so called discrete variable representation (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 expansion
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 singlemode displacement level. Additionally, collectivedisplacement should (optimally) minimize the interaction between modes in the PESmanybody expansion approximation. Ideally, pairterms 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 describe the fundamental transitions of lowlying 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 Figure 2.1, for the PES of a methyl deformation (rotational) mode of Nmethylacetamide(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 ith 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, consequently 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 massweighted 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 coordinates. 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 coordinates 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 representation 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 ith 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 apriori knowledge of the potential, choosing an appropriate range is not a simple 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 minimum becomes sufficiently small (low frequency modes), the induced displacementscan become important, as previously shown in Figure 2.1. Thus, grids should preferably 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 difficulties that might occur when the resulting atomic displacements are becoming tooimportant.
In this work, we obtained modeoptimal grids by first evaluating the single mode potentials at the HartreeFock (HF) level employing a double or triple zeta basis set withfew points on a sufficiently large grid. After interpolation on a denser grid, an appropriate 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 semiempirical 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 secondorder 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 respective 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 error due to the effective nature of the interactions between the oscillators. Again,following electronic structure theory, it is possible to correct this by including vibrational correlation effects in VSCF using either second order perturbation theory(VMP2) [70], coupledcluster (VCC) [71, 72], configurationinteraction (VCI) [73,74] or a multiconfigurational SCF (VMCSCF) and its extension completeactivespace 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 several 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 zerothoder 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)MP2m =
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 viceversa, 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 regular 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 perturbation 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 asVMP2m.
2.4.2 Vibrational Configuration Interaction
When two or more strongly mixed states of a coupled oscillator are accidentally degenerate 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 HartreeProduct, 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 spacesize 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 Hamiltonian 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 introduced 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 configurations 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 onedimensional 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 numerically 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 GAMESSUS [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 pairpotentials. Therefore, it is possible toeasily and efficiently add, remove or replace potentials. Further computational efficiency can be achieved by noticing that the integral Vnm can be precomputed (numericalcontraction) in the outerloop 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 offdiagonal 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 offdiagonal 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 offdiagonal 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 jnii 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 jknii
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 jnii 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 jknii
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 jknii
n jj nkk
17 r e t u r n Hnm
2.5 AbInitio Potentials and VSCF 25
2.5 AbInitio 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 abinitio 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 accepted by the chemistry community as a costeffective 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 augccpVTZ basis set [87]. The latter is capable ofan accuracy of 1 kJ/mol, for relative conformational energies.
On the other hand, postHF methods like MP2 and CCSD(T) represent an established hierarchy to approach the exact solution of the electronic Schrodinger equation.Unfortunately, the delocalized character of the canonical HartreeFock orbitals, traditionally 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 likeDensityFitting (DF), also known as ResolutionoftheIdentity (RI) [88, 89], wherethe fourcenter twoelectron integrals are replaced by linear combinations of threecenter 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 AbInitio 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 (DFLMP2) with analytical gradients [93] has been made available in the Molpro program [94] recently. Asshown by Hrenar and al. [95] the impact of DFLMP2 on the harmonic vibrational frequencies is negligible. This is an important remark, as the DFLMP2 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 geometry, which can be problematic for local correlation methods as the local orbital domains 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 DFMP2 approach over the DFLMP2.
2.5.2 MultiLevel Calculations, Model Potentials andCoupling Norms
It has been noted earlier that while the computation of the harmonic transition frequencies and loworder diagonal anharmonicities requires sophisticated abinitio methodsto reach near spectroscopic accuracy, it is possible to obtain reasonably accurate intermode coupling potentials from simpler methods like semiempirical or even chargeinteraction models [29]. This fact has been used widely in the simulation of multidimensional IR spectra where the experimentally more easily accessible transition frequencies have been used in combination with relatively simple models for the description of the mode couplings.
The hierarchical potential energy expansion in Eq. (2.7) allows for a straightforwardimplementation of such a multilevel 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 abinitio calculations with coupling potentials V (2)i j (Qi,Q j)from a lower level computation. This multilevel scheme can of course also be applied
2.5 AbInitio Potentials and VSCF 27
to higher order coupling potentials [98]. In this work multilevel computations will beidentified by a shorthand notation of the form D:P:T where D describes the methodused to obtain the diagonal potential P the paircoupling potentials and T the triplepotentials respectively.
Due to the quickly increasing number of PES points required in the standard approach, the applicability of VSCF to peptides and proteins is currently limited to PESexpansions based on classical forcefields. Even using a multiple level scheme, accurate abinitio 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. Nevertheless, in order to be amenable for the simulation of vibrational spectra of larger peptidesand proteins, the generation of abinitio 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 paircoupling 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
~r2
)n ni=1
x2i , F(0) = 0 (2.25)
2.5 AbInitio 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~r3 + d~r4,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 potential
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 nonnegligible reduction of a factorof 16.
The potentials in Figures 2.4 and 2.5 for the the paircouplings 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 inclusion of higher order terms in the polynomial approximation. Also, in the case of
2.5 AbInitio Potentials and VSCF 29
Figure 2.4: NMA PES plot for paircoupling potential of amideI/II, model (left) and full PES (right)
Figure 2.5: NMA PES plot for paircoupling potential of modes1/3, model (left) and full (right)
2.5 AbInitio Potentials and VSCF 30
vibrational modes with bending or torsional character couplings are typically overestimated as the model potential does not decay quickly enough to zero when approachingthe coordinate axis compared to the true potential.
Without any apriori knowledge of the coupling potentials and their importance forthe spectrum, any approach that depends on the selection of coupling potentials requires 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 intramolecular 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 onedimensionalsubproblems
(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 integrals 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 onedimensionalwave function (1)i . In this approximation only four evaluations of the coupling potential 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, interactive, objectoriented programming language. Although a VSCF algorithm is freelyavailable [101] through the GAMESSUS [82] program, using a program written ina highlevel language such as Python offers many advantages. Perhaps the most important ones with respect to efficiency are the possibility to do coarsegrain parallelprocessing of the PES grid points by interfacing to any external program that can evaluate such points as well as a efficient and flexible CI implementation. Thanks to theversatility offered by a highlevel scripting language, tailoring very specific CI Hamiltonians 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 onedimensional vibrational 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 converged within less than 1 cm1 for all states shown in Table 2.1. For the VMP2 andVMP2m perturbation approaches we used a maximum excitation level of 5. Threedifferent VSCFVCI computations with no restriction on the type of excitations (i.e.up to hexupleexcitations) 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 sufficiently 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 paircoupling 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 between vibrational modes. Here, vibrational modes are numbered from lowest to highestfrequency, which corresponds to CH2 wagging, CH2 asymmetric bend, CH2 symmetric 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 obtained 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 wavefunction with ni = 6 and no restriction on Nmax.
State Nr State EV SCF EVMP2corr EVMP2mcorr 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 VMP2m 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 VMP2m corrections are on average closer tothe VSCFCI results.
1 2 3 4 5 6
1
2
3
4
5
6
Figure 2.6: Formaldehyde paircouplings V (2)i j and correspondingnorms mapping. Upper left triangle: Pairpotentialterms. 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 maximum norm has been scaled to 1 and the color codingon the interval [0,1] ranges from green to red.
Our large VSCFCI computation (see Table 2.1(c)) reproduces the results of both Romanowski [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 attributed 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 accuracy of approximatively one wavenumber compared to the large VSCFCI 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 VSCFCI computations in Table 2.1(b) do not display a faster convergence 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 statespecific methods witha pure modeexcitation 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 converged to within less than 2 cm1. The SS results show also the same pattern of convergence 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 fundamentals. 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
917
.52
918
.57
818
.57
818
.89
618
.89
618
.89
618
.89
618
.89
618
.89
62
1,0,
0,0,
0,0
49
.02
549
.63
952
.37
552
.77
853
.37
353
.75
953
.38
453
.77
053
.38
453
.77
03
0,1,
0,0,
0,0
7.
290
7.
0424
24
.56
624
.15
325
.32
224
.89
625
.57
225
.13
725
.57
225
.13
74
0,0,
1,0,
0,0
8.
171
8.
3296
26
.41
325
.93
127
.45
926
.94
227
.56
127
.04
627
.56
127
.04
65
0,0,
0,1,
0,0
6.
486
5.
9080
22
.98
921
.51
024
.13
822
.66
124
.51
822
.98
924
.51
822
.98
96
0,0,
0,0,
1,0
18
.23
470
.24
0223
.83
415
1.31
030
.30
516
6.07
30
.30
516
6.07
30
.30
516
6.07
47
0,0,
0,0,
0,1
6.
735
6.
5606
25
.72
124
.81
126
.68
525
.68
026
.68
525
.68
026
.68
525
.68
08
2,0,
0,0,
0,0
19
.11
494
.18
2092
.96
913
8.82
096
.24
318
0.88
996
.25
718
1.04
096
.25
718
1.04
09
0,2,
0,0,
0,0
11
.60
110
.96
0632
.19
531
.09
633
.31
332
.15
133
.58
132
.39
833
.58
132
.39
810
0,0,
2,0,
0,0
11
.27
011
.89
1834
.24
733
.49
836
.57
035
.67
936
.71
035
.82
736
.71
035
.82
711
0,0,
0,2,
0,0
23
.89
519
.00
3941
.18
734
.68
042
.47
935
.94
742
.93
736
.27
342
.93
736
.27
312
0,0,
0,0,
2,0
98.47
913
1.82
5723
2.42
611
7.55
412
4.45
815
4.75
112
4.45
815
4.75
112
4.45
815
4.75
113
0,0,
0,0,
0,2
16
.38
914
.17
8840
.59
837
.11
743
.44
539
.33
643
.44
539
.33
643
.44
539
.33
614
1,1,
0,0,
0,0
24
.52
826
.98
4756
.35
256
.50
959
.19
459
.08
759
.82
559
.71
059
.82
559
.71
015
1,0,
1,0,
0,0
22
.90
123
.32
9360
.35
857
.93
563
.63
560
.88
564
.09
561
.32
764
.09
561
.32
716
0,1,
1,0,
0,0
13
.57
514
.41
0919
.52
319
.06
638
.67
437
.33
839
.44
738
.06
339
.44
738
.06
317
1,0,
0,1,
0,0
3.
238
8.
8606
36
.51
837
.79
139
.09
340
.14
440
.10
841
.11
540
.12
141
.12
618
0,1,
0,1,
0,0
0.40
66.
9918
15
.21
813
.97
832
.43
330
.24
933
.27
531
.06
933
.57
231
.31
819
0,0,
1,1,
0,0
3.86
35.
4141
15
.18
914
.04
933
.57
030
.91
034
.76
431
.96
734
.89
532
.07
220
1,0,
0,0,
1,0
94
.88
916
1.43
015
5.28
120
4.25
916
7.61
123
2.07
416
7.71
223
2.50
916
7.71
223
2.50
921
0,1,
0,0,
1,0
35.99
917
.15
638
.52
486
.50
743
.35
815
4.95
548
.96
916
7.49
948
.97
116
7.50
122
0,0,
1,0,
1,0
111.
102
38
.54
04.
914
14
4.35
87.
476
16
3.80
07.
735
16
3.97
37.
735
16
3.97
323
0,0,
0,1,
1,0
51.98
75.
861
17.00
170
.30
01.
102
12
5.96
92.
936
13
6.92
510
.42
613
6.92
824
1,0,
0,0,
0,1
11
.96
428
.25
047
.97
748
.92
752
.30
452
.80
852
.37
052
.86
852
.37
052
.86
825
0,1,
0,0,
0,1
7.17
01.
083
8.
379
8.
288
26
.07
125
.11
626
.74
225
.65
126
.74
225
.65
126
0,0,
1,0,
0,1
39
.25
043
.79
755
.77
652
.08
968
.61
464
.78
168
.92
365
.04
168
.92
365
.04
127
0,0,
0,1,
0,1
11.05
72.
308
12
.67
211
.50
531
.57
529
.12
232
.70
229
.99
432
.70
229
.99
428
0,0,
0,0,
1,1
75.80
967
.75
50.
006
17
5.92
313
.34
219
1.15
313
.38
919
2.34
213
.38
919
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 confirmed the importance of quadruples in the VCI expansion for obtaining reliable fundamental 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 1520 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, thioformamide [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 suspected to be involved in the ultrafast relaxation of the amideI 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 abinitio 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 analysis where carried out using the Molpro [94] program at the MP2/augccpVTZ levelwithout symmetry constraints, resulting in planar structures that are true minima. Theanharmonic frequency computation using vibrational second order perturbation theory (see Section 1.2.2) was carried out using the Gaussian 03 [41] program at theMP2/augccpVTZ level with tightly optimized geometries. The differences in the optimized 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 using Molpro at the DFMP2/augccpVTZ and CCSD(T)/augccpVTZ level for thediagonals V (1)i , at the DFMP2/augccpVTZ for the pairs V
(2)i j , and finally at the DF
MP2/ccpVTZ 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 using splines. Vibrationalrotational 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/augccpVTZ 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 unrealistic 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 semiquartic forcefield representation of the PES. It is rather a fortuitous cancellation of errors in the nonvariational 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 semiquartic PT2 forcefield are presented, the RMSDvalue of the variational method is larger (39.6 cm1) than for PT2. Adding correlationeffects shows that the VMP2m 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 VMP2m 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 inplaness symmetric stretch opb outofplane 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