Distributed Atomic Polarizabilities of Amino Acids and their Hydrogen-Bonded Aggregates

51 Pages·2016·1.1 MB·English

Distributed Atomic Polarizabilities of Amino Acids
and their Hydrogen-Bonded Aggregates
Leonardo H. R. Dos Santos,*,† Anna Krawczuk,‡ and Piero Macchi*,†
† Department of Chemistry and Biochemistry, University of Bern, Freiestrasse 3, 3012 Bern,
Switzerland
‡ Faculty of Chemistry, Jagiellonian University, Ingardena 3, 30-060 Kraków, Poland
ABSTRACT
With the purpose of rational design of optical materials, distributed atomic polarizabilities of
amino acid molecules and their hydrogen-bonded aggregates are calculated in order to identify
the most efficient functional groups, able to buildup larger electric susceptibilities in crystals.
Moreover, we carefully analyze how the atomic polarizabilities depend on the one-electron basis
set or the many-electron Hamiltonian, including both wave function and density functional
theory methods. This is useful for selecting the level of theory that best combines high accuracy
and low computational costs, very important in particular when using the cluster method to
estimate susceptibilities of molecular-based materials.
KEYWORDS: distributed polarizability, quantum theory of atoms in molecules, electron density
distribution, linear optical properties, hydrogen bonding.

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1. INTRODUCTION
The response of a molecule to an electric field has received enormous attention from
computational chemists, due to the important implications for opto-electronic properties of
materials.1-3 The linear and non-linear optical responses are correlated to the dipole electric
(hyper)polarizabilities and susceptibilities at molecular and macroscopic levels, respectively.4
With the current computational resources, (hyper)polarizabilities of medium-size molecules in
the gas-phase can be computed with satisfactory accuracy if sufficient electronic correlation is
included and extended atomic basis sets are used. This has enabled the design of molecular-
based optical materials, but, as stated by Champagne and Bishop,5 the knowledge acquired in the
field of single-molecule (hyper)polarizability calculations must evolve to quantitatively
rationalize optical properties of the solid state, a field which is far less advanced. In fact, with
few exceptions, a useful electro-optical material will be in the solid state and, frequently, in a
crystalline or partially crystalline phase. For this reason, the ultimate goal is the investigation of
optical properties in periodically homogeneous systems.
Crystal-orbital or plane-wave-based calculations are in principle the correct approaches to model
crystalline effects. However, some problems affect these methods: a) the amount of electronic
correlation that one can introduce is limited;6 b) convergence often fails when crystal-orbital
Bloch-type wave functions contain diffuse atomic functions; c) plane-wave calculations exclude
the localized core-orbital functions and therefore their contribution to the (hyper)polarizabilities.
In order to overcome these drawbacks, some approaches have emerged, for example the so-
called supermolecule or cluster method.5 Within this approach, the (hyper)polarizabilities of
several interacting molecules are evaluated as a whole, just like in standard molecular
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calculations. The (hyper)polarizabilities of a molecule embedded in a crystal are then estimated
as average quantities over the cluster, and eventually used to approximate the electric
susceptibilities of a periodic system. By comparing the (hyper)polarizabilities of an isolated
molecule and that extracted from the molecular cluster, one gains insight into the role of short
and medium-range intermolecular interactions, crucial for the design of optical materials.7
Because density functional theory (DFT) methods may be inappropriate for correct evaluation of
the intermolecular dispersion forces, one of the goals of the present work is to identify the most
accurate DFT functionals able to estimate optical properties in molecular assemblies, using
calculations of high-level electron correlation as benchmarks. The performance of DFT have
been rigorously investigated for the polarizabilities of large water clusters.8 It was found that the
accuracy of post-Hartree-Fock levels of theory was not sufficiently higher than that of DFT to
justify the increased computational costs. In this work, we aim at extend this analysis to amino
acid aggregates, through a careful estimation of the cost/benefits and the marginal utility of
extending the level of theory.
A subject that has also attracted attention is the breakdown of the molecular polarizability into
atomic contributions.9 Visualizing and analyzing atomic and bond polarizabilities of a system is
useful because functional groups represent the way in which chemists reduce molecules for
synthetic and engineering purposes. In fact, a given molecular property may especially originate
from one particular group. In this respect, the transferability of atomic and functional group
properties is a key concept which allows exporting quantities, calculated with high accuracy in
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small reference molecules, to atoms or functional groups belonging to complex systems, like
macromolecules, polymers or crystals,10-12 that would be too expensive to calculate ab initio.
Several schemes have been proposed for the calculation of distributed atomic polarizabilities.13-21
Bader and co-workers22 adopted the topological partitioning of the electron density within the
Quantum Theory of Atoms in Molecules (QTAIM). The method has been successfully applied
for the evaluation of intermolecular interaction energies23-25 and the transferability of
electrostatic properties in n-alkanes.26 Subsequently, Keith27 generalized Bader’s method, within
the “atomic response theory”, recently modified and implemented in a program (PolaBer) by
some of us.28 The main advantage is the removal of origin-dependent terms, which are
particularly disadvantageous for the transferability of atomic or group quantities. PolaBer
extends the quantities computed from the atomic polarizabilities, including bond polarizabilities
and evaluation of refractive indices in crystals.
In fact, QTAIM offers several advantages, in particular because it leads to an exact partitioning
of the molecular electron density in real-space, in contrast with the Hilbert-space schemes such
as the one developed by Stone29,30 in the framework of the distributed multipole analysis.31
Recently, the fuzzy Hirshfeld partitioning scheme of the electron density has also been applied to
extract molecular polarizabilities in clusters32,33 or atomic polarizabilities in large molecular
systems.9 However, a disadvantage of the Hirshfeld scheme, at least in the current
implementations, is that the functional-group polarizabilities remain origin-dependent and,
therefore, cannot be properly exported to other systems.
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In the present work, the QTAIM distributed atomic first-order dipole polarizabilities of amino
acids and their hydrogen-bonded aggregates are calculated at different theoretical levels. We
focused on the first-order polarizability tensor (cid:2745) due to its relative simplicity, its major role in
both qualitative and quantitative considerations of reaction paths and molecular interactions, and,
of course, its prominence in determining linear optical properties of materials. For this analysis,
we have chosen the natural -amino acids in their zwitterionic form, because the optical
properties of their crystals, co-crystals, salts and metal hybrids have attracted much attention in
the last few years.34
While some components of the total polarizability tensors are measurable, either in the gas or in
the condensed phase,35 no procedure is available for the experimental determination of
polarizabilities of atoms in molecules. Therefore, for benchmarking the various DFT functionals,
we use post-Hartree-Fock methods up to coupled-cluster (CC) techniques, known to deliver
highly accurate polarizabilities.8
Another purpose of our work is testing the efficiency of Gaussian-type basis sets for the
calculations of distributed polarizabilities. Both Pople and Dunning families of basis sets are
considered. These investigations allow us to propose a protocol for the evaluation of optical
properties of molecular materials that is both quality- and cost-oriented.
The last goal of our study is to quantify the perturbation of hydrogen bonds on the molecular
polarizabilities, analyzing the most common aggregation modes of amino acids in the solid state.
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2. THEORETICAL METHODS
2.1. Electron density calculations
Electron densities were obtained by molecular-orbital wave function calculations at various
levels of approximation. We used the glycine molecule and three hydrogen bonded dimers as
references in order to test various basis set expansions and to compare results from different DFT
functionals against those from configuration interaction (CI) or perturbation methods (second-
order Møller-Plesset, MP2). Firstly, the polarizabilities of the glycine monomer and the dimers
were investigated with increasing level of electron correlation, up to coupled-cluster singles-and-
doubles (CCSD), and increasing basis-set rank, up to quadruple-zeta quality including diffuse
and polarization functions. Secondly, the polarizability of the glycine monomer was calculated
using various density functionals and then compared against results at the highest electron-
correlated levels. Thirdly, results with basis sets from Pople and Dunning families were
compared against each other at the CAM-B3LYP level of theory, found to be one of the best-
performing functional. Finally, all the 20 amino acids and some dimers were investigated at the
CAM-B3LYP/d-aug-cc-pVDZ level, which was determined to be the best compromise between
accuracy and computational costs.
When available, three-dimensional coordinates were taken from measured single crystal neutron
diffraction data and kept frozen. This choice is necessary because the most stable configuration
of the amino acids in the gas-phase is the neutral one, whereas the zwitterionic forms are the
most frequently observed species in liquid solutions and crystal-phases. Because stationary
points for the zwitterionic forms are normally not found in gas-phase potential energy surfaces,
we could not use optimized geometries. For those amino acids in which X-ray structures only are
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available, the heavier atom positions were kept at measured values and the distances to the
attached H atoms were normalized to the average neutron diffraction values, as given by the
International Tables for Crystallography. Because the crystal structure of lysine is not known, the
calculations were performed for its cationic form, lysinium. Correlated calculations use the
frozen-core approximation. All the wave-function calculations were performed using the
Gaussian 09 package36 and the corresponding charge-density distributions were partitioned in
keeping with the QTAIM using the AIMAll program.37
2.2. Distributed polarizability calculations
The PolaBer program28 was used to calculate distributed atomic polarizabilities. The method is
based on QTAIM partitioning of the ground-state and field-perturbed electron densities (cid:2025)(cid:4666)(cid:2200)(cid:4667) of a
molecular system into its atomic contributions. The algorithm was already tested38 and proved a
reliable way to breakdown the molecular (cid:2745) tensor into atomic contributions. Because the
partitioning scheme is “exact”, the sum of the atomic polarizabilities must coincide, a part for
minor numerical imprecision, to the molecular polarizability calculated in Gaussian 09 using the
coupled-perturbed Hartree-Fock (CPHF) or Kohn-Sham (CPKS) equations.
The QTAIM-based topological analysis allows one to represent the total molecular dipole
moment as a sum of atomic components (cid:2246)(cid:4666)Ω(cid:4667), where Ω is the atomic basin-volume. Each atomic
dipole moment is the sum of a polarization (cid:2246) (cid:4666)Ω(cid:4667) and a charge translation (cid:2246) (cid:4666)Ω(cid:4667) contribution:10
(cid:3043) (cid:3030)
(cid:2246)(cid:4666)Ω(cid:4667) (cid:3404) (cid:2246) (cid:4666)Ω(cid:4667)(cid:3397)(cid:2246) (cid:4666)Ω(cid:4667) (cid:3404) (cid:3398)(cid:1516) (cid:4670)(cid:2200)(cid:3398)(cid:2174) (cid:4671)(cid:2025)(cid:4666)(cid:2200)(cid:4667)(cid:1856)(cid:2200)(cid:3397)(cid:4670)(cid:2174) (cid:3398)(cid:2174) (cid:4671)(cid:1869)(cid:4666)Ω(cid:4667) (1)
(cid:3043) (cid:3030) Ω Ω Ω (cid:2868)
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Where (cid:1869)(cid:4666)Ω(cid:4667) is the net charge of the atom Ω, (cid:2174) is the position vector of Ω and (cid:2174) is the
Ω (cid:2868)
arbitrary origin of the molecular coordinate system. Although the origin-dependent charge-
translation contribution can be converted to an alternative, origin-independent definition,22 only
recently a computationally convenient method for such calculation has been proposed.27 It was
slightly modified and implemented by some of us in the PolaBer software.28
The static first-order dipole polarizability tensor of a molecular system, (cid:2745), is the gradient of the
electric dipole moment (cid:2246) with respect to an external, uniform, and time-independent electric
field (cid:2161):
(cid:2246) (cid:2246) (cid:2246)
(cid:2235) (cid:3404) (cid:2191)(cid:4672) (cid:4673)(cid:3397)(cid:2192)(cid:3436) (cid:3440)(cid:3397)(cid:2193)(cid:4672) (cid:4673) (2)
(cid:3006)(cid:3299) (cid:3006)(cid:3300) (cid:3006)(cid:3301)
As for the dipole moment, the molecular polarizability tensor can be decomposed into additive
atomic contributions:
(cid:2235) (cid:3404) ∑(cid:3015)(cid:3276) (cid:2235)(cid:4666)Ω(cid:4667) (cid:3404)∑(cid:3015)(cid:3276) (cid:3427)(cid:2235) (cid:4666)Ω(cid:4667)(cid:3397)(cid:2235) (cid:4666)Ω(cid:4667)(cid:3431) (3)
Ω(cid:2880)(cid:2869) Ω(cid:2880)(cid:2869) (cid:3043) (cid:3030)
Where (cid:1840) is the total number of atoms in the molecular system. (cid:2235) (cid:4666)Ω(cid:4667) and (cid:2235) (cid:4666)Ω(cid:4667) are the
(cid:3028) (cid:3043) (cid:3030)
polarization and charge translation atomic polarizability tensors, respectively. These terms arise
from the derivation of the corresponding (cid:2246) (cid:4666)Ω(cid:4667) and (cid:2246) (cid:4666)Ω(cid:4667) atomic dipole moments respect to the
(cid:3043) (cid:3030)
applied electric field, according to Eqn. 2.
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Nuclear relaxations can also affect the computed linear and non-linear optical properties,39,40
however they are not considered in this work, meaning that we focus only on the high-frequency
electron polarizability. In further work, we will investigate this aspect as well.
Given the linear response of the electron density with respect to a sufficiently small applied
field,27 (cid:2235)(cid:4666)Ω(cid:4667) can be calculated by numerical differentiation, using wave functions computed at
finite electric fields. Thus, the atomic polarizability components (cid:2009) (cid:4666)Ω(cid:4667) are evaluated as:
(cid:3036)(cid:3037)
(cid:3091)(cid:3254)(cid:3285)(cid:4666)Ω(cid:4667)(cid:2879)(cid:3091)(cid:3116)(cid:4666)Ω(cid:4667)
(cid:2009) (cid:4666)Ω(cid:4667) (cid:3404) lim (cid:3284) (cid:3284) (4)
(cid:3036)(cid:3037)
(cid:2161)→(cid:2868) (cid:3006)(cid:3285)
Where (cid:2020)(cid:3006)(cid:3285)(cid:4666)Ω(cid:4667) is the dipole moment component of the atomic basin Ω along the (cid:1861) direction
(cid:3036)
computed with an applied electric field in direction (cid:1862). Eqn. 4 does not take into account the
coupling between atomic volume and atomic charge. For this reason, the atomic polarizability
tensors result slightly asymmetric and require tensor symmetrization.41
Atomic and molecular polarizability tensors can be visualized as ellipsoids in the same three-
dimensional space as the molecule, assuming 1 Å(cid:2871) ≡ 1 Å. In this work, a scaling factor of
0.4 Å(cid:2879)(cid:2870) is applied to reduce the size of polarizability ellipsoids for visualization purposes.
All the amino acids under study and their aggregates have (cid:1829) symmetry. Therefore, the first-
(cid:2869)
order polarizability tensors have six different non-zero components. We report the components
of the diagonalized tensors ((cid:2009) , (cid:2009) , (cid:2009) ). The isotropic polarizability was estimated as:
(cid:2869)(cid:2869) (cid:2870)(cid:2870) (cid:2871)(cid:2871)
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(cid:2869)
(cid:2009) (cid:3404) (cid:4666) (cid:2009) (cid:3397)(cid:2009) (cid:3397)(cid:2009) (cid:4667) (5)
(cid:3010)(cid:3020)(cid:3016) (cid:2869)(cid:2869) (cid:2870)(cid:2870) (cid:2871)(cid:2871)
(cid:2871)
The anisotropy of the polarizability tensor is typically estimated by ∆(cid:2009):35
(cid:2869)/(cid:2870)
∆(cid:2009) (cid:3404) (cid:4676)(cid:2869)(cid:4670)3Tr(cid:4666)(cid:2745)(cid:2870)(cid:4667)(cid:3398)(cid:4666)Tr(cid:2745)(cid:4667)(cid:2870)(cid:4671)(cid:4677) (6)
(cid:2870)
All polarizabilities are reported in atomic units unless stated otherwise. Sources of error on the
calculated polarizabilities are due to the incompleteness of both electronic correlation models
and basis sets, and the limited accuracy of the atomic basin integration procedures.42
3. RESULTS AND DISCUSSION
3.1. Atomic and functional group polarizabilities in amino acid molecules
One of the main aims of our work is quantifying the contribution of individual atoms and, more
importantly, of functional groups to specific optical properties, like the refractive index.
Distributed atomic polarizabilities are particularly useful, because they enable reconstructing the
polarizability of a functional group in a molecule by simple sum of the atomic tensors. Fig. 1
shows the polarizabilities for the twenty -amino acids, in their zwitterionic configuration,
calculated at the CAM-B3LYP/d-aug-cc-pVDZ level of theory. As already discussed,38 atomic
polarizabilities are extremely sensitive to the local chemical environment, being larger along the
directions of covalently bonded atoms. For example, in the carboxylic groups, the polarizability
ellipsoids of the oxygen atoms are stretched in the direction of the C–O bonds, because these
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