Efficient model chemistries for peptides. II.
Basis set convergence in the B3LYP method. Pablo ECHENIQUE
Instituto de Biocomputación y Física de Sistemas Complejos (BIFI),
and Departamento de Física Teórica, Universidad de Zaragoza,
Pedro Cerbuna 12, E-50009 Zaragoza, Spain
E-mail<EMAIL_ADDRESS>Gregory A. CHASS
Global Institute Of COmputational Molecular and Materials Science (GIOCOMMS),
and School of Chemistry, University of Wales, Bangor, Gwynedd, LL57 2UW United
Kingdom,
and College of Chemistry, Beijing Normal University, Beijing, 100875, China

PACS: 07.05.Tp; 31.15.Ar; 31.50.Bc; 87.14.Ee; 87.15.Aa; 89.75.-k

Keywords: peptides, quantum chemistry, PES, B3LYP, basis set convergence

Abstract

Small peptides are model molecules for the amino acid residues that are the
constituents of proteins. In any bottom-up approach to understand the
properties of these macromolecules essential in the functioning of every
living being, to correctly describe the conformational behaviour of small
peptides constitutes an unavoidable first step. In this work, we present an
study of several potential energy surfaces (PESs) of the model dipeptide HCO-
L-Ala-NH2. The PESs are calculated using the B3LYP density-functional theory
(DFT) method, with Dunning’s basis sets cc-pVDZ, aug-cc-pVDZ, cc-pVTZ, aug-cc-
pVTZ, and cc-pVQZ. These calculations, whose cost amounts to approximately 10
years of computer time, allow us to study the basis set convergence of the
B3LYP method for this model peptide. Also, we compare the B3LYP PESs to a
previous computation at the MP2/6-311++G(2df,2pd) level, in order to assess
their accuracy with respect to a higher level reference. All data sets have
been analyzed according to a general framework which can be extended to other
complex problems and which captures the nearness concept in the space of model
chemistries (MCs).

##  1 Introduction

In any bottom-up attempt to understand the behaviour of protein molecules (in
particular, the still elusive protein folding process [3, 1, 5, 2, 4]), the
characterization of the conformational preferences of short peptides [13, 12,
7, 11, 6, 9, 10, 8] constitutes an unavoidable first step. Due to the lower
numerical effort required and also to the manageability of their
conformational space, the most frequently studied peptides are the shortest
ones: the _dipeptides_ [14, 17, 16, 15], in which a single amino acid residue
is capped at both the N- and C-termini with neutral peptide groups. Among
them, the most popular choice has been the _alanine_ dipeptide [34, 30, 26,
23, 27, 24, 21, 22, 6, 20, 29, 19, 33, 25, 31, 28, 32, 18], which, being the
simplest chiral residue, shares many similarities with most of the rest of
dipeptides for the minimum computational price.

Although classical force fields [35, 36, 37, 38, 39, 40, 41, 42, 43] are the
only feasible choice for simulating large molecules at present, they have been
reported to yield inaccurate _potential energy surfaces_ (PESs) for dipeptides
[44, 45, 46, 47, 29] and short peptides [48, 6]. Therefore, it is not
surprising that they are widely recognized as being unable to correctly
describe the intricacies of the whole protein folding process [49, 50, 51, 44,
52, 53, 54, 55]. On the other hand, albeit prohibitively demanding in terms of
computational resources, ab initio quantum mechanical calculations [56, 57,
58] are not only regarded as the correct physical description that in the long
run will be the preferred choice to directly tackle proteins (given the
exponential growth of computer power and the advances in the search for
pleasantly scaling algorithms [60, 59]), but they are also used in small
peptides as the reference against which less accurate methods must be compared
[61, 62, 44, 45, 47, 29, 6] in order to, for example, parameterize improved
generations of additive, classical force fields for polypeptides.

However, despite the sound theoretical basis, in practical quantum chemistry
calculations a plethora of approximations must be typically made if one wants
to obtain the final results in a reasonable human time. The exact ‘recipe’
that includes all the assumptions and steps needed to calculate the relevant
observables for any molecular system has been termed _model chemistry_ (MC) by
John Pople. In his own words, a MC is an “approximate but well-defined general
and continuous mathematical procedure of simulation” [63].

After assuming that the particles involved move at non-relativistic velocities
and that the greater weight of the nuclei allows to perform the Born-
Oppenheimer approximation, we are left with the problem of solving the non-
relativistic electronic Schrödinger equation [60]. The two starting
approximations to its exact solution that a MC must contain are, first, the
truncation of the $N$-electron space (in wavefunction-based methods) or the
choice of the functional (in DFT) and, second, the truncation of the one-
electron space, via the LCAO scheme (in both cases). The extent up to which
the first truncation is carried (or the functional chosen in the case of DFT)
is commonly called the _method_ and it is denoted by acronyms such as RHF,
MP2, B3LYP, CCSD(T), FCI, etc., whereas the second truncation is embodied in
the definition of a finite set of atom-centered Gaussian functions termed
_basis set_ [60, 64, 57, 58, 65], which is also designated by conventional
short names, such as 6-31+G(d), TZP or cc-pVTZ(–f). If we denote the method by
a capital $M$ and the basis set by a $B$, the specification of both is
conventionally denoted by $L:=M/B$ and called a _level of the theory_. Typical
examples of this are RHF/3-21G or MP2/cc-pVDZ [56, 57, 58].

Note that, apart from these approximations, which are the most commonly used
and the only ones that are considered in this work, the MC concept may include
a lot of additional features: the heterolevel approximation (explored in a
previous work in this series [34]), protocols for extrapolating to the
infinite-basis set limit [66, 67, 68, 69, 70], additivity assumptions [71, 72,
73, 74], extrapolations of the Møller-Plesset series to infinite order [75],
removal of the so-called _basis set superposition error_ (BSSE) [76, 77, 78,
79, 80, 81, 82], etc. The reason behind most of these techniques being the
urging need to reduce the computational cost of the calculations.

Now, although general applicability is a requirement that all MCs must
satisfy, general accuracy is not mandatory. Actually, the fact is that the
different procedures that conform a given MC are typically parameterized and
tested in very particular systems, which are often small molecules. Therefore,
the validity of the approximations outside that native range of problems must
be always questioned and checked. However, while the approximate computational
cost of a given MC for a particular system is rather easy to predict on the
basis of simple scaling relations, its expected accuracy on a particular
problem could be difficult to predict a priori, specially if we are dealing
with large molecules in which interactions in very different energy scales are
playing a role. The description of the conformational behaviour of peptides
(or, more generally, flexible organic species), via their PESs in terms of the
soft internal coordinates, is one of such problems and the one that is treated
in this work.

To this end, we first describe, in sec. 2, the computational and theoretical
methods used throughout the rest of the document. Then, in sec. 3, we
introduce a basic framework that rationalizes the actual process of evaluating
the efficiency of any MC for a complex problem. These general ideas are used,
in sec. 4, to perform an study of the _density-functional theory_ (DFT) B3LYP
[83, 84, 85, 86] method with the cc-pVDZ, aug-cc-pVDZ, cc-pVTZ, aug-cc-pVTZ,
and cc-pVQZ Dunning’s basis sets [87, 88]. To this end, we apply these levels
of the theory to the calculation the PES of the model dipeptide HCO-L-Ala-NH2
(see fig. 1), and assess their efficiency by comparison with a reference PES.
Finally, in sec. 5, the most important conclusions are briefly summarized.

##  2 Methods

All ab initio quantum mechanical calculations have been performed using the
GAMESS-US program [89, 90] under Linux and on 2.2 GHz PowerPC 970FX machines
with 2 GB RAM memory.

The internal coordinates used for the Z-matrix of the HCO-L-Ala-NH2 dipeptide
in the GAMESS-US input files are the _Systematic Approximately Separable
Modular Internal Coordinates_ (SASMIC) ones introduced in ref. 91. They are
presented in table 1 (see also fig. 1 for reference).

Atom name | Bond length | Bond angle | Dihedral angle
---|---|---|---
H1 |  |  |
C2 | (2,1) |  |
N3 | (3,2) | (3,2,1) |
O4 | (4,2) | (4,2,1) | (4,2,1,3)
C5 | (5,3) | (5,3,2) | (5,3,2,1)
H6 | (6,3) | (6,3,2) | (6,3,2,5)
C7 | (7,5) | (7,5,3) |  $\phi:=$(7,5,3,2)
C8 | (8,5) | (8,5,3) | (8,5,3,7)
H9 | (9,5) | (9,5,3) | (9,5,3,7)
H10 | (10,8) | (10,8,5) | (10,8,5,3)
H11 | (11,8) | (11,8,5) | (11,8,5,10)
H12 | (12,8) | (12,8,5) | (12,8,5,10)
N13 | (13,7) | (13,7,5) |  $\psi:=$(13,7,5,3)
O14 | (14,7) | (14,7,5) | (14,7,5,13)
H15 | (15,13) | (15,13,7) | (15,13,7,5)
H16 | (16,13) | (16,13,7) | (16,13,7,15)
Table 1: Internal coordinates in Z-matrix form of the protected dipeptide HCO-
L-Ala-NH2 according to the SASMIC scheme introduced in ref. 91. The numbering
of the atoms is that in fig. 1, and the soft Ramachandran angles $\phi$ and
$\psi$ are indicated.

All PESs in this study have been discretized into a regular 12$\times$12 grid
in the bidimensional space spanned by the Ramachandran angles $\phi$ and
$\psi$, with both of them ranging from $-165^{\mathrm{o}}$ to
$165^{\mathrm{o}}$ in steps of $30^{\mathrm{o}}$. To calculate the PES at a
particular level of the theory, we have run constrained energy optimizations
at each point of the grid, freezing the two Ramachandran angles $\phi$ and
$\psi$ at the corresponding values. In order to save computational resources,
the starting structures were taken, when possible, from PESs previously
optimized at a lower level of the theory. All the basis sets used in the study
have been taken from the GAMESS-US internally stored library, and spherical
Gaussian-type orbitals (GTOs) have been preferred, thus having 5 d-type and 7
f-type functions per shell.

Figure 1:  Atom numeration of the protected dipeptide HCO-L-Ala-NH2 according
to the SASMIC scheme introduced in ref. 91. The soft Ramachandran angles
$\phi$ and $\psi$ are also indicated.

We have computed 5 PESs, using the DFT B3LYP [83, 84, 85, 86] method with the
cc-pVDZ, aug-cc-pVDZ, cc-pVTZ, aug-cc-pVTZ, and cc-pVQZ Dunning’s basis sets
[87, 88]. The total cost of these calculations in the machines used is around
10 years of computer time.

Also, let us note that the correcting terms to the PES coming from mass-metric
tensors determinants and from the determinant of the Hessian matrix have been
recently shown to be relevant for the conformational behaviour of peptides
[18]. (The latter may be regarded as a residual entropy arising from the
elimination of the hard coordinates from the description.) Although, in this
study, we have included none of these terms, the PES calculated here is the
greatest part of the effective free energy [18], so that it may be considered
as the first ingredient for a further refinement of the study in which the
correcting terms are taken into account. The same may be said about another
important source of error in the calculation of relatives energies in peptide
systems: the already mentioned BSSE [31].

In order to compare the PESs produced by the different MCs, a statistical
criterium (distance) introduced in ref. 92 has been used. Let us recall here
that this _distance_ , denoted by $d_{12}$, profits from the complex nature of
the problem studied to compare any two different potential energy functions,
$V_{1}$ and $V_{2}$. From a working set of conformations (in this case, the
144 points of each PES), it statistically measures the typical error that one
makes in the _energy differences_ if $V_{2}$ is used instead of the more
accurate $V_{1}$, admitting a linear rescaling and a shift in the energy
reference.

Despite having energy units, the quantity $d_{12}$ approximately presents all
properties characteristic of a typical mathematical metric in the space of MCs
(hence the word ‘distance’), such as the possibility of defining a symmetric
version of it and a fulfillment of the triangle inequality (see ref. 92 for
the technical details and sec. 3 for more about the importance of these
facts). It also presents better properties than other quantities customarily
used to perform these comparisons, such as the energy RMSD, the average energy
error, etc., and it may be related to the Pearson’s correlation coefficient
$r_{12}$ by

$d_{12}=\sqrt{2}\,{\sigma}_{2}(1-r_{12}^{2})^{1/2}\ ,$ (1)

where $\sigma_{2}$ is the standard deviation of $V_{2}$ in the working set.

Moreover, due to its physical meaning, it has been argued in ref. 92 that, if
the distance between two different approximations of the energy of the same
system is less than $RT$, one may safely substitute one by the other without
altering the relevant dynamical or thermodynamical behaviour. Consequently, we
shall present the results in units of $RT$ (at $300^{\mathrm{o}}$ K, so that
$RT\simeq 0.6$ kcal/mol).

Finally, if one assumes that the effective energies compared will be used to
construct a polypeptide potential and that it will be designed as simply the
sum of mono-residue ones (more complex situations may be found in real
problems [93]), then, the number $N_{\mathrm{res}}$ of residues up to which
one may go keeping the distance $d_{12}$ between the two approximations of the
the $N$-residue potential below $RT$ is [92]

$N_{\mathrm{res}}=\left(\frac{RT}{d_{12}}\right)^{2}\ .$ (2)

According to the value taken by $N_{\mathrm{res}}$ for a comparison between a
fixed reference PES, denoted by $V_{1}$, and a candidate approximation,
denoted by $V_{2}$, we shall divide the whole accuracy range in sec. 4 in
three regions depending on the accuracy: the _protein region_ , corresponding
to $0<d_{12}\leq 0.1RT$, or, equivalently, to $100\leq
N_{\mathrm{res}}<\infty$; the _peptide region_ , corresponding to
$0.1RT<d_{12}\leq RT$, or $1\leq N_{\mathrm{res}}<100$; and, finally, the
_inaccurate region_ , where $d_{12}>RT$, and even for a dipeptide it is not
advisable to use $V_{2}$ as an approximation to $V_{1}$. Of course, these are
only approximate regions based on the general idea that we are not interested
on the dipeptides as a final system, but only as a mean to approach protein
behaviour from the botton-up. Therefore, not only the error in the dipeptides
must be measured, but it must also be estimated how this discrepancy
propagates to polypeptide systems.

##  3 General framework

The general abstract framework behind the investigation presented in this
study (and also implicitly behind most of the works found in the literature),
may be described as follows:

The objects of study are the _model chemistries_ defined by Pople [63] and
discussed in the introduction. The MCs under scrutiny are applied to a
particular _problem_ of interest, which may be thought to be formed by three
ingredients: the _physical system_ , the _relevant observables_ and the
_target accuracy_. The MCs are then selected according to their ability to
yield numerical values of the relevant observables for the physical system
studied within the target accuracy. The concrete numerical values that one
wants to approach are those given by the _exact model chemistry_ MCε, which
could be thought to be either the experimental data or the exact solution of
the non-relativistic electronic Schrödinger equation [60]. However, the
computational effort needed to perform the calculations required by MCε is
literally infinite, so that, in practice, one is forced to work with a
_reference model chemistry_ MCref, which, albeit different from MCε, is
thought to be close to it. Finally, the set of MCs that one wants to
investigate are compared to MCref and the nearness to it is seen as
approximating the nearness to MCε.

Figure 2:  Space $\mathcal{M}$ of all model chemistries. The exact model
chemistry MCε is shown as a black circle, the MP2 reference MC is shown as a
grey-filled circle, and B3LYP MCs as white-filled ones. Both reference PESs
are indicated with an additional circle around the points. The situation
depicted is (schematically) the one found in this study, assuming that MP2 is
a more accurate method than B3LYP to account for the conformational
preferences of peptide systems. The positions of the different MCs have no
relevance, and only the relative measured distances among them are
qualitatively depicted.

These comparisons are commonly performed using a numerical quantity
$\mathcal{D}$ that is a function of the relevant observables. In order for the
intuitive ideas about relative proximity in the $\mathcal{M}$ space to be
captured and the above reasoning to be meaningful, this numerical quantity
$\mathcal{D}$ must have some of the properties of a mathematical distance. In
particular, it is advisable that the _triangle inequality_ is obeyed, so that,
for any model chemistry MC, one has that

$\displaystyle\mathcal{D}(\mathrm{MC}_{\varepsilon},\mathrm{MC})\leq\mathcal{D}(\mathrm{MC}_{\varepsilon},\mathrm{MC}^{\mathrm{ref}})+\mathcal{D}(\mathrm{MC}^{\mathrm{ref}},\mathrm{MC})\
,$ (3a)
$\displaystyle\mathcal{D}(\mathrm{MC}_{\varepsilon},\mathrm{MC})\geq\big{|}\mathcal{D}(\mathrm{MC}_{\varepsilon},\mathrm{MC}^{\mathrm{ref}})-\mathcal{D}(\mathrm{MC}^{\mathrm{ref}},\mathrm{MC})\big{|}\
,$ (3b)

and, assuming that
$\mathcal{D}(\mathrm{MC}_{\varepsilon},\mathrm{MC}^{\mathrm{ref}})$ is small
(and $\mathcal{D}$ is a positive function), we obtain

$\mathcal{D}(\mathrm{MC}_{\varepsilon},\mathrm{MC})\simeq\mathcal{D}(\mathrm{MC}^{\mathrm{ref}},\mathrm{MC})\
,$ (4)

which is the sought result in agreement with the ideas stated at the beginning
of this section.

The distance $d_{12}$ introduced in ref. 92 and summarized in the previous
section, measured in this case on the conformational energy surfaces (the
relevant observable) of the model dipeptide HCO-L-Ala-NH2 (the physical
system), approximately fulfills the triangle inequality and thus captures the
_nearness_ concept in the space $\mathcal{M}$ of model chemistries.

This space, $\mathcal{M}$, containing all possible MCs, is a rather complex
and multidimensional one. For example, two commonly used ‘dimensions’ which
may be thought to parameterize $\mathcal{M}$ are the size of the basis set and
the amount of electron correlation in the model (or the quality of the DFT
functional used). However, since there are many ways in which the size of a
basis set or the electron correlation may be increased and there are
additional approximations that can be included in the MC definition (see sec.
1), the ‘dimensions’ of $\mathcal{M}$ can be considered to be many more than
two.

The definition of a distance, such as the one described in the previous lines,
for a given problem of interest helps to provide a certain degree of structure
into this complex space. In fig. 2 a two-dimensional scheme of the overall
situation found in this study is presented.

##  4 Results

MCs |  $d_{12}/RT$ a |  $a_{12}$ b |  $N_{\mathrm{res}}$ c |  $t$ d
---|---|---|---|---
B3LYP/aug-cc-pVTZ | 0.079 | 15.2 | 159.8 | 79.09%
B3LYP/cc-pVTZ | 0.191 | 21.1 | 27.4 | 9.78%
B3LYP/aug-cc-pVDZ | 0.172 | 82.8 | 33.7 | 5.27%
B3LYP/cc-pVDZ | 1.045 | 109.4 | 0.9 | 1.29%
Table 2: Basis set convergence results for the B3LYP MCs investigated in this
work. aDistance with the B3LYP/cc-pVQZ reference in units of $RT$ at
$300^{\mathrm{o}}$ K. bEnergy offset with the reference MC in kcal/mol.
cMaximum number of residues in a polypeptide potential up to which the
corresponding MC may correctly approximate the reference (under the
assumptions in sec. 2). dRequired computer time, expressed as a fraction of
$t_{\mathrm{ref}}$.

Before starting with the results of the calculations, let us introduce the
concept of _efficiency_ of a particular MC that shall be used: It is laxly
defined as a balance between accuracy (in terms of the distance introduced in
sec. 2) and computational cost (in terms of computer time). It can be
graphically extracted from the _efficiency plots_ , where the distance
$d_{12}$ between any given MC and a reference one is shown in units of $RT$ in
the $x$-axis, while, in the $y$-axis, one can find the computer time taken for
each MC (see the following pages for two examples). As a general thumb-rule,
_we shall consider a MC to be more efficient for approximating the reference
when it is placed closer to the origin of coordinates in the efficiency plot_.
This approach is intentionally non-rigorous due to the fact that many factors
exist that influence the computer time but may vary from one practical
calculation to another; such as the algorithms used, the actual details of the
computers (frequency of the processor, size of the RAM and cache memories,
system bus and disk access velocity, operating system, mathematical libraries,
etc.), the starting guesses for the SCF orbitals or the starting structures in
geometry optimizations.

Taking all this into account, the only conclusions that shall be drawn in this
work about the relative efficiency of the MCs studied are those deduced from
strong signals in the plots and, therefore, those that can be extrapolated to
future calculations; in other words, _the small details shall be typically
neglected_.

Figure 3: Efficiency plot of all the B3LYP MCs studied. In the $x$-axis, we
show the distance $d_{12}$, in units of $RT$ at $300^{\mathrm{o}}$ K, between
any given MC and the B3LYP/cc-pVQZ reference (indicated by an encircled
point), while, in the $y$-axis, we present the computer time needed to compute
the whole 12$\times$12 grid in the Ramachandran space of the model dipeptide
HCO-L-Ala-NH2. The different accuracy regions are labeled, and the 10% of the
time $t_{\mathrm{best}}$ taken by the reference MC is also indicated.

In the first part of the study, we compare all B3LYP MCs to the one with the
largest basis set, B3LYP/cc-pVQZ (the highest level of the theory calculated
for this work, depicted in fig. 4) using the distance introduced in sec. 2.
All mentions to the accuracy of any given MC in this part must be understood
as relative to this reference. However, it has been reported that MP2 is a
superior method to B3LYP to account for the conformational behaviour of
peptide systems [94]. Therefore, the absolute accuracy of the B3LYP MCs
calculated here is probably closer to the relative accuracy with respect to
the MP2/6-311++G(2df,2pd) reference in what follows. In this spirit, this part
of the study should be regarded as an investigation of the convergence to _the
infinite basis set B3LYP limit_ , for which the best B3LYP MC here is probably
a good approximation.

Figure 4:  Potential energy surface of the model dipeptide HCO-L-Ala-NH2
computed at the B3LYP/cc-pVQZ level of the theory. The PES has been originally
calculated in a 12$\times$12 discrete grid in the space spanned by the
Ramachandran angles $\phi$ and $\psi$ and later smoothed with bicubic splines
for visual convenience. The energy reference has been set to zero. (At this
level of the theory, the absolute energy of the minimum point in the
12$\times$12 grid, located at $(-75^{o},75^{o})$, is $-417.199231353$
hartree).

The results are depicted in fig. 3, and in table 2. We can extract several
conclusions from them:

  * •

Regarding the convergence to the infinite basis set limit, we observe that
only the most expensive MC, B3LYP/aug-cc-pVTZ, correctly approximates the
reference for peptides of more than 100 residues. On the other hand, for only
5.27% of the computer time $t_{\mathrm{ref}}$ taken by the reference MC, we
can use B3LYP/aug-cc-pVDZ, which correctly approximates it up to 30-residue
peptides. Finally, the MC with the smallest basis set, B3LYP/cc-pVDZ cannot
properly replace the reference even in dipeptides.

  * •

In ref. [34], using Pople’s basis sets [95, 96, 102, 97, 98, 99, 100, 101], we
saw that “the general rule that is sometimes assumed when performing quantum
chemical calculations, which states that ‘the more expensive, the more
accurate’, is rather coarse-grained and relevant deviations from it may be
found.” We recognized that “One may argue that this observation is due to the
unsystematic way in which Pople basis sets can be enlarged and that the
correlation between accuracy and cost will be much higher if, for example,
only Dunning basis sets are used.”, which is definitely observed in fig. 3,
but we argued that this was something to be expected, since “there are two few
Dunning basis sets below a reasonable upper bound on the number of elements to
see anything but a line in the efficiency plot”. In the results presented in
this work, we can see that, even if the correlation between accuracy and cost
is higher in the case of Dunning’s basis sets than in the case of Pople’s, due
to the smaller number of the former, we can still observe that the thumb-rule
‘the more expensive, the more accurate’ breaks also in this case, since the
B3LYP/aug-cc-pVDZ MC is, at the same time, more accurate and less costly than
B3LYP/cc-pVTZ. In general, this idea applies to all the approximations that a
MC may contain (see the introduction for a partial list), and justifies the
systematic search for the most efficient combination of them for a given
problem. This work is our second step (ref. [34] is the first one) in that
path for the particular case of the conformational behaviour of peptide
systems.

  * •

The observation in the previous point also suggests that it may be efficient
to include diffuse functions (the ‘aug-’ in aug-cc-pVDZ) in the basis set for
this type of problems.

  * •

The error of the studied MCs regarding the differences of energy (as measured
by $d_{12}$) is much smaller than the error in the absolute energies (as
measured by $a_{12}$), suggesting that the largest part of the discrepancy
must be a systematic one.

In the second part of the study, we assess the absolute accuracy of the B3LYP
MCs by comparing them to the (as far as we are aware) highest homolevel in the
literature, the MP2/6-311++ G(2df,2pd) PES in ref. [34]. If one assumes that
this level of the theory may be close enough to the exact result for the given
problem at hand, then this comparison measures the ‘absolute’ accuracy of the
B3LYP MCs, and not only their relative accuracy with respect to the B3LYP
infinite basis set limit, as we did in the previous part. This is the
fundamental difference between figs. 3 and 5.

MCs |  $d_{12}/RT$ a |  $a_{12}$ b |  $N_{\mathrm{res}}$ c |  $t$ d
---|---|---|---|---
B3LYP/cc-pVQZ | 1.008 | -457.2 | 0.98 | 1861
B3LYP/aug-cc-pVTZ | 1.029 | -442.0 | 0.94 | 1472
B3LYP/cc-pVTZ | 1.058 | -436.1 | 0.89 | 182
B3LYP/aug-cc-pVDZ | 1.006 | -374.4 | 0.99 | 98
B3LYP/cc-pVDZ | 1.533 | -347.8 | 0.43 | 24
Table 3: Comparison of all the B3LYP MCs investigated in this work with the
MP2/6-311++G(2df,2pd) in ref. 34. aDistance with the MP2/6-311++G(2df,2pd)
reference in units of $RT$ at $300^{\mathrm{o}}$ K. bEnergy offset with the
reference MC in kcal/mol. cMaximum number of residues in a polypeptide
potential up to which the corresponding MC may correctly approximate the
reference (under the assumptions in sec. 2). dComputer time needed for the
calculation of the whole PES, in days.

The results of this part of the study are depicted in fig. 5, and in table 3.
We can extract several conclusions from them:

  * •

All B3LYP MCs, including the largest one, B3LYP/cc-pVQZ, lie in the inaccurate
region of the efficiency plot in fig. 5, meaning that they cannot be reliably
used to approximate the MP2/6-311++G(2df,2pd) reference even in the smallest
dipeptides.

  * •

Related with the observations in the previous part of the study, we see that
there is no point, if one is worried about absolute accuracy, in going beyond
the aug-cc-pVDZ basis set in B3LYP.

  * •

The B3LYP/cc-pVDZ MC again performs significantly worse than the rest,
agreeing with the results in the previous part of the study, and suggesting
that cc-pVDZ may be a too small basis set for the problem tackled here.

  * •

Again, the error of the MCs in the differences of energy (as measured by
$d_{12}$) is much smaller than the error in the absolute energies (as measured
by $a_{12}$).

Figure 5: Efficiency plot of all the B3LYP MCs studied. In the $x$-axis, we
show the distance $d_{12}$, in units of $RT$ at $300^{\mathrm{o}}$ K, between
any given MC and the MP2/6-311++G(2df,2pd) reference calculated in ref. 34,
while, in the $y$-axis, we present the computer time needed to compute the
whole 12$\times$12 grid in the Ramachandran space of the model dipeptide HCO-
L-Ala-NH2. The different accuracy regions are labeled

##  5 Conclusions

In this study, we have investigated 5 PESs of the model dipeptide HCO-L-Ala-
NH2, calculated with the B3LYP method, and the cc-pVDZ, aug-cc-pVDZ, cc-pVTZ,
aug-cc-pVTZ, and cc-pVQZ Dunning’s basis sets. We have first assessed the
convergence of the B3LYP MCs to the infinite basis set limit, and then we have
evaluated their absolute accuracy by comparing them to the (as far as we are
aware) highest homolevel in the literature, the MP2/6-311++G(2df,2pd) PES in
ref. [34]. All the comparisons have been performed according to a general
framework which is extensible to further studies, and using a distance between
the different PESs that correctly captures the nearness concept in the space
of MCs. The calculations performed here have taken around 10 years of computer
time.

The main conclusions of the study are the following:

  * •

The complexity of the problem (the conformational behaviour of peptides)
renders the correlation between accuracy and computational cost of the
different quantum mechanical algorithms imperfect. This ultimately justifies
the need for systematic studies, such as the one presented here, in which the
most efficient MCs are sought for the particular problem of interest.

  * •

Assuming that the MP2/6-311++G(2df,2pd) level of the theory is closer to the
exact solution of the non-relativistic electronic Schrödinger equation than
B3LYP/cc-pVQZ, B3LYP is not a reliable method to study the conformational
behaviour of peptides. Even if, as we emphasize at the end of this section, it
may be dangerous to state that a method that performs well in the particular
model of an alanine residue studied here will also be recommendable for longer
and more complex peptides, we can clearly _reject_ any method that already
fails in HCO-L-Ala-NH2.

  * •

If B3LYP is still needed to be used, due to, for example, computational
constraints, aug-cc-pVDZ represents a good compromise between accuracy and
cost.

  * •

The error of the studied MCs regarding the differences of energy (as measured
by $d_{12}$) is much smaller than the error in the absolute energies (as
measured by $a_{12}$), suggesting that the largest part of the discrepancy
must be a systematic one.

Finally, let us stress again that the investigation performed here have used
one of the simplest dipeptides. The fact that we have treated it as an
isolated system, the small size of its side chain and also its aliphatic
character, all play a role in the results obtained. Hence, for bulkier
residues included in polypeptides, and, specially for those that contain
aromatic groups, those that are charged or may participate in hydrogen-bonds,
the methods that have proved to be efficient here must be re-tested and the
conclusions drawn about the B3LYP convergence to the infinite basis set limit,
as well as those regarding the comparison between B3LYP and MP2, should be re-
evaluated.

## Acknowledgments

The numerical calculations in this work have been performed thanks to a
computer time grant at the Zaragoza node (Caesaraugusta) of the Spanish
Supercomputing Network (RES). We thank all the support staff there, for the
efficiency at solving the problems encountered. We also thank J. L. Alonso for
illuminating discussions.

This work has been supported by the research projects DGA (Aragón Government,
Spain) E24/3 and MEC (Spain) FIS2006-12781-C02-01. P. Echenique is supported
by a MEC (Spain) postdoctoral contract.

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