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J. Phys. Chem. B 2008, 112, 1540-1548
Guanine Crystals: A First Principles Study F. Ortmann,* K. Hannewald, and F. Bechstedt European Theoretical Spectroscopy Facility (ETSF), Institut fu¨r Festko¨rpertheorie und -optik, Friedrich-Schiller-UniVersita¨t, Max-Wien-Platz 1, 07743 Jena, Germany ReceiVed: August 11, 2007; In Final Form: October 4, 2007
We report first principles density functional theory studies on the basic ground state characteristics, dynamic properties, and the electronic structure of guanine crystals. The effect of water molecules within the crystal is studied in detail, and we discuss their influence on the structural, vibrational, and electronic properties. The geometries calculated for various crystal structures are compared with gas-phase calculations and available experimental data. Phonon frequencies and eigenvectors are predicted for intermolecular and intramolecular lattice vibrations. Vibrational and electronic density-of-states are presented and analyzed. The electronic band structure near the fundamental gap is calculated from the Kohn-Sham approach. We find that the former molecular HOMO states form a dispersive band in the π-π stacking direction upon condensation resulting in a large bandwidth of 0.83 eV. Consequences for the charge transport in layered van der Waals bonded organic molecular crystals are discussed.
I. Introduction Helical nanowires of DNA molecules have been regarded for some time as passive molecules with the sole purpose to store genetic information. Nowadays, they are also considered as promising candidates for molecular devices toward a further miniaturization of electronic technology. The electric properties of such wires have been actively studied in the past few years.1-8 This includes efforts with a biochemical motivation, since understanding electronic transport through DNA is essential to characterize and control important life processes, such as radiation damage and repair.9-11 Heterocyclic DNA bases differ in their recognition properties from homocyclic molecules such as pentacene which has already been used as a transistor material.12,13 A directional bond due to strong hydrogen bridges between adjacent bases controls the self-assembly process, and may be used to build up devices in a bottom-up approach. Among the DNA base molecules, guanine is in the focus of the activities because of its low ionization potential which should favor charge transport.14 Fourstranded quasi one-dimensional DNA structures known as quadruplexes or DNA tetraplexes show interesting chemicophysical and biological aspects.15-17 Quadruple helical structures of homoguanilic or guanine-rich sequences occur in a wide variety of natural situations and organisms. In addition, selfassembled two-dimensional layers of guanosine ribbons (see ref 18 and references therein) interconnecting source and drain electrodes may be used to study electronic transport. Guanosine and its derivatives are of interest because of their peculiar sequence of hydrogen bond donor or acceptor groups facilitating the self-organization process. Furthermore a three-dimensional crystal structure is formed in the guanine monohydrate crystal.19 In this case, the guanine molecules form planes of rings consisting of six molecules in a hexagonal fashion. Columns of water molecules pass through the interstices of these hexagonal arrays and stabilize the layered crystal. * Corresponding author.
There is an obvious correlation between supramolecular ordering and conduction properties.18 This has already been exploited successfully in single-crystal devices based on wellknown materials like rubrene or pentacene.12,20 Similarly, also three-dimensional agglomerates of DNA bases such as molecular crystals of guanine monohydrate may possess interesting electronic properties and should attract attention in the field of organic electronics in addition to the standard materials rubrene and pentacene. However, details of the guanine structures and their consequences for the (bio)physical and (bio)chemical properties are less known and understood. Much computational effort has been spent to determine the structural properties of the guanine molecule as well as its vibronic and electronic excitations. But much less is found for condensed aggregates.21,22 One reason is the complexity of the problem to describe simultaneously the weak intermolecular bonds and the strong intramolecular covalent bonds. In this respect, a prototypical example is the guanine monohydrate crystal (cf. Figures 1 and 2). Apart from general bonding issues, the importance of the water columns for the stability of this crystal is not well-known. This includes the number of H2O molecules, their positions, and their consequences for the electronic structure and charge-carrier transport. The temperature dependence of the latter depends significantly on the inter- and intramolecular lattice vibrations. Their frequencies and coupling strengths are essentially unknown. The purpose of the present paper is therefore twofold. First, the computational studies of guanine monohydrate crystals presented here give novel insight into the structural, vibrational, and electronic properties of these particular organic molecular crystals. Hereby, special attention is paid to the so far unknown influence of water columns and their relevance for the abovementioned properties. Second, from a somewhat broader perspective, our calculations serve as a prototypical study of three-dimensional crystals built up from DNA base molecules. Here, we put a special focus on the relevance of our findings for the charge-carrier transport in such crystals.
10.1021/jp076455t CCC: $40.75 © 2008 American Chemical Society Published on Web 01/16/2008
Guanine Crystals
Figure 1. Monoclinic unit cell of the guanine monohydrate crystal with atomic basis of four guanine molecules. Neighboring parallel layer and water columns along the c axis are indicated.
Figure 2. Crystal structure of the guanine monohydrate crystal projected down the c axis. The dashed box indicates the unit cell with the lattice vector b. Black double lines represent hydrogen atoms in a hydrogen bridge bond of the OH‚‚‚O type. The atoms in one guanine molecule are labeled.
The paper is organized as follows. In section II, we describe our computational apparatus to study guanine crystals on an ab initio level. The results from these calculations are presented in section III. In a first step, we investigate the crystal structure of guanine monohydrate for various geometries and compare the results to available experimental data. Second, we study intra- and intermolecular vibrational properties of the system. Finally, the electronic structure of the crystal is examined in detail. In particular, we discuss the formation of bands with surprisingly large dispersions and the resulting consequences for electron and hole transport. The paper is concluded by a short summary in section IV. II. Computational Methods A. Total Energy and Structure. We simulate the total energy of the crystal within the framework of density functional theory (DFT). The explicit calculations are performed using the VASP package.23,24 We apply local and semi-local approximations to the exchange-correlation (XC) functional as discussed below. The projector augmented wave scheme25 is used for the simulation of the interaction of valence electrons and cores. The basis set for the wavefunctions in the interstitial region consists of plane waves with an energy cutoff of 37 Ry. For a variety of molecular systems with first-row elements, this approach has been proven to sufficiently describe ground-state and vibrational properties.14,26-28 According to the assumed monoclinic crystal structure displayed in Figure 1, the Brillouin zone sampling is
J. Phys. Chem. B, Vol. 112, No. 5, 2008 1541 performed using a regular grid of k points with large dimensions of 2 × 3 × 10. Forces on the atoms are calculated as HellmannFeynman forces. In the structural relaxation steps, the threshold for residual forces is set to a very small value of 0.1 meV/Å. The atomic positions with smaller residual forces are considered as the equilibrium structure. For the crystal structure, we follow the experimental results from X-ray diffraction19 suggesting a monoclinic crystal with axes a, b, and c and the monoclinic angle β between a and c (see Figure 1). They give rise to a starting geometry for the total energy minimization with a unit cell containing four guanine molecules. As shown in Figures 1 and 2, these molecules arrange in a more or less planar configuration in (3h01) planes. The normal vector to the (3h01) planes, however, is not exactly perpendicular to the molecular planes. Figure 1 shows the tilt of one pair of guanine molecules with respect to another. As a result, the molecular normal vectors are tilted with respect to each other and with respect to the normal vector of the (3h01) planes as well. Water molecules are arranged in columnar structures interconnecting these planes. Laterally, the arrangement is stabilized by a hydrogen-bonding network between the guanine molecules and between the water columns and the guanine molecules. This bonding is different from what has been proposed for a guanine monolayer on graphite29 or the Hoogsteen-bonded guanine tetraplexes.30 From Figure 2, it can be seen that each water molecule forms an OH‚‚‚O hydrogen bond to an oxygen atom of one guanine molecule as well as an NH‚‚‚O hydrogen bond with an amino group of another guanine molecule. In the experimental studies,19 the positions of the water hydrogens could not be determined, but also the positions of the water oxygens along the H2O columns are affected by considerable uncertainty. In addition, the stoichiometry of the real guanine monohydrate crystal may differ from that of the ideal structure presented in Figures 1 and 2. Nevertheless, the dominating structure is clearly identified as a stacking of guanine molecules hydrogen-bonded to the water columns with space-group symmetry P21/c (C52h) and associated point group C2h. In the original paper, Thewalt et al. (cf. ref 19) discussed that the water columns may have a more complicated structure. This is motivated since the space group symmetry P21/c suggests symmetric hydrogen bonds that is, that the water hydrogens not hydrogen-bonded to the guanines are in a symmetric position between the water oxygens which appears as an unlikely geometry. A more realistic asymmetric assignment of these hydrogens lowers the symmetry to either P21 (C22) or Pc (C2S) with associated point groups C2 and CS, respectively. For better visualization, these three possible structures of the H2O columns are sketched in Figure 3 in a view along the monoclinic b axis. The assignment of the hydrogens to the upper oxygens in both columns gives the CS structure whereas the assignment to upper oxygens in one column and to lower oxygens in the other column results in a crystal with C2 symmetry. For that reason, we investigate the three different monohydrate structures and compare energetic results, ground-state geometries, and vibrational and electronic properties. Henceforth, they will be named according to their point group symmetry: the C2h, C2, and CS structure. Additionally, the gas-phase guanine molecule is simulated as well in order to supplement the analysis of the crystals with reference data. These calculations are performed using a large cubic unit cell of length 24 Å in order to minimize the interaction of the guanine molecule with its images.
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Figure 3. Schematic drawing of the columnar H2O structures in the high-symmetry geometry (top) and low-symmetry configurations (middle and bottom).
B. Approximation for Exchange and Correlation Functional. Since the exact XC functional of DFT is not known, one has to choose an approximate form. There is a variety of different approximations which can be classified as local/ semilocal functionals and hybrid functionals. Among the local/ semilocal ones, one can further distinguish between the local density approximation (LDA)31 and the semilocal gradientcorrected density functionals in the generalized gradient approximation (GGA).32 GGA functionals perform better for hydrogen-bonded systems such as water or ice33,34 as well as for intramolecular hydrogen bonds in amino acids.26 However, they fail in describing electron correlations originating from the van der Waals (vdW) interaction.35,36 Similarly, hybrid functionals such as B3LYP or HSE03 do not account for vdW forces either. In the case of guanine monohydrate crystals, both types of interaction, the vdW interaction and hydrogen bonding, play important roles. The interaction between the layers formed by guanine molecules is mainly of the vdW type, while the intralayer bonding is dominated by hydrogen mediated bridges. Hence, we would like to perform our calculations using an XC functional that can describe these interactions equally well. Unfortunately, such an XC functional is not known, and we have to choose one out of the pool of different approaches mentioned above. From the DFT treatment of layered crystals such as graphite,35,36 we know that the LDA functional gives reasonable values for atomic distances and still acceptable values for the lattice vibrations. For that reason, we choose the DFTLDA treatement as a pragmatic but accurate way to describe the ground state and dynamic properties of DNA-base crystals. This LDA-based concept has already been employed successfully in the case of other organic molecular crystals, including oligo-acenes37 and durene.28 Technically, the LDA-XC energy is calculated according to the parametrization of Perdew and Zunger.38 For comparison, we also carry out test calculations with GGA gradient corrections employing the PW91 functional.39 However, the hybrid functionals have no additional value for the simulation of ground
Ortmann et al. state properties at least not at their computational costs and will therefore not be considered in the present work. C. Dynamic and Electronic Properties. The lattice vibrations are described within the harmonic approximation. We calculate the force constant matrix in a central difference scheme and set up a generalized eigenvalue problem of rank 228, that is, including all intra- and intermolecular modes of the guanine molecules and the water columns. A symmetrization procedure restores the symmetry of the modes that might be broken in the scheme due to numerical errors. Because of the large unit cells, we restrict ourselves to the Γ point in the Brillouin zone. The resulting eigenfrequencies allow for the calculation of the vibrational density of states (DOS) using a Gaussian broadening of 10 cm-1. A detailed description of the computational procedure for the lattice-vibrational properties can be found in ref 40. The electronic properties are derived from the Kohn-Sham (KS) eigenvalues of the Bloch states. These single-particle eigenvalues are generally not related to any single-electron excitation energy in the system since quasiparticle effects are neglected. As a consequence of the lack of the excitation aspect, all energy gaps and transition energies are underestimated.41 This holds in particular for the fundamental gap for which KS values are about 30-100% smaller than experimental measured ones. On the other hand, KS energy bands exhibit a k dispersion that is indeed comparable to those of measured Bloch bands. In order to approach the true quasiparticle gap, we address this problem within a “delta self-consistent field” (∆SCF) scheme.42 Since we study molecular crystals, we take the quasiparticle gap corrections as those calculated for the molecule. This procedure improves the value for the gap energy of the crystal; however, it tends to overestimate it. A more detailed description of this method can be found in ref 28. We mention here that it works very well for both single-particle and two-particle excitation energies of localized electronic systems, such as molecules.14,27 Since the dispersion of the KS bands is in reasonable agreement with measurements, quantities related to the band dispersion like the effective mass tensor of a semiconductor band or the bandwidth, are often discussed in terms of the KS approach. We follow this strategy throughout this paper. Thereby, temperature effects are omitted. The temperature limitations can be overcome within an advanced theory taking the electron-phonon interaction into account.37 Moreover, such a theory, based on a generalized Holstein model, also allows to explicitly calculate the mobility tensors of electrons and holes as well as their temperature dependences.43 However, this is not the scope of the present paper. III. Results and Discussion A. Atomic Geometry and Crystal. The lattice parameters, three lattice constants a, b, and c as well as the monoclinic angle β between the axes a and c (see Figure 1), derived from the total energy minimization are listed in Table 1 for the three different local arrangements of the H2O columns. The DFTLDA results are compared with the results derived within the DFT-GGA scheme and measured data from ref 19. For the LDA lattice constants, we obtain equilibrium values that are below those found experimentally. For a, b, and c, this underestimation amounts to 5.4%, 3.3%, and 2.9%, respectively. These deviations seem to be in agreement with the general overbinding tendency for covalent bonds using a local approach to XC. On the other hand, the deviation of the monoclinic angle β from its measured value is smaller. Within the DFT-GGA
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TABLE 1: Lattice Constants a, b, and c (Angstroms) and Monoclinic Angle β (Degrees) for the Guanine Monohydrate Crystal in Different Structures and for Different XC Approximations with Experimental Reference Data from Ref 19 LDA a b c β
GGA
C2h
CS
C2
C2h
exptl
15.62 10.91 3.54 94.2
15.68 10.93 3.53 94.4
15.68 10.93 3.53 94.4
16.35 11.25 4.024 98.9
16.510(8) 11.277(8) 3.645(5) 96.8(1)
TABLE 2: Intramolecular Bond Lengths (Angstroms) of the Guanine Molecules in the Guanine Monohydrate Crystal and in Gas Phase with Experimental Data from Ref 19 LDA bond
C2h
CS/C2
gas phase
exptl
N1-C2 N1-C6 C2-N2 C2-N3 N3-C4 C4-C5 C4-N9 C5-C6 C5-N7 C6-O6 N7-C8 C8-N9 H1-N2 H2-N2 H3-N1 H4-C8 H5-N9
1.365 1.367 1.322 1.335 1.337 1.393 1.358 1.405 1.370 1.263 1.317 1.358 1.037 1.045 1.099 1.092 1.081
1.365 1.367 1.322 1.336 1.337 1.393 1.358 1.404 1.370 1.263 1.316 1.358 1.037 1.045 1.099 1.092 1.080
1.359 1.422 1.356 1.307 1.342 1.394 1.360 1.427 1.366 1.220 1.304 1.373 1.017 1.015 1.023 1.090 1.018
1.371 1.398 1.333 1.315 1.364 1.392 1.364 1.405 1.405 1.239 1.319 1.369 1.030 1.009 1.029 1.020 1.012
description, the two largest lattice constants, a and b, are increased and approach the measured values. This can be explained with the correction of the overbinding of the hydrogen-bonding network of the guanines in agreement with general observations for hydrogen bridge bonds (cf. refs 26, 33, 34). In order to explain the strong elongation of the c lattice constant within GGA, one has to consider the bonding mechanism perpendicular to the guanine sheets. Bonding in c direction is only partially governed by the hydrogen bonds of the water columns. Instead, the vdW interaction is expected to play a substantial role in the bonding along the c direction. As a result, within GGA, one observes a significant expansion of the lattice perpendicular to the guanine sheets. The indirect guanine-water-guanine bridge obviously cannot prevent the expansion of the lattice in c direction that is caused by the failure of GGA describing the vdW interaction. The GGA calculations give intermolecular plane spacings of 3.64 Å. Compared with the experimental value of 3.30 Å,19 this overestimation (10%) is larger than the underestimation in LDA (3.15 Å or -4.5%). Finally, we compare the different local hydrogen configurations. As seen from Table 1, one observes only very small changes in the lattice parameters going from the high-symmetry H2O columns to columns of lower symmetry. We conclude from Table 1 that the influence of the detailed arrangement of the hydrogen atoms in the columns on the lattice and, hence, on the intermolecular spacing of the guanine molecules is negligibly small. Between the two low-symmetry structures, there is practically no difference at all. Intramolecular bond lengths and bond angles for the guanine molecule are collected in Tables 2 and 3 . The labels used here are defined in Figure 2. For comparison, the tables show experimental data taken from ref 19 as well. Calculated structural parameters are given for the three water column
TABLE 3: Intramolecular Bond Angles (Degrees) of Guanine Molecules in the Guanine Monohydrate Crystal with Experimental Data from Ref 19 LDA angle
C2h
CS/C2
gas phase
exptl
N1-C2-N2 N1-C2-N3 N1-C6-C5 N1-C6-O6 C2-N1-C6 C2-N3-C4 N2-C2-N3 N3-C4-C5 N3-C4-N9 C4-C5-C6 C4-C5-N7 C4-N9-C8 C5-C4-N9 C5-C6-O6 C5-N7-C8 C6-C5-N7 N7-C8-N9 H1-N2-C2 H1-N2-H2 H2-N2-C2 H3-N1-C2 H3-N1-C6 H4-C8-N7 H4-C8-N9 H5-N9-C4 H5-N9-C8
117.7 122.2 113.9 119.8 124.8 114.4 120.1 126.6 127.5 118.1 109.6 106.7 105.9 126.3 105.0 132.3 112.7 120.5 115.7 123.5 119.2 115.7 125.2 122.1 124.6 128.6
117.8 122.2 113.9 119.9 124.8 114.4 120.1 126.6 127.5 118.1 109.6 106.7 105.9 126.2 105.0 132.3 112.7 120.1 116.0 123.5 119.0 115.9 125.3 122.0 124.8 128.4
117.6 123.0 109.7 119.0 126.9 112.9 119.4 129.1 126.1 118.4 110.7 107.0 104.8 131.3 105.0 130.9 112.4 115.1 117.4 120.5 120.7 112.4 125.7 121.9 125.2 127.8
115.3 124.6 111.9 120.4 124.6 111.9 120.0 127.6 126.2 119.2 109.6 107.0 106.1 127.7 104.2 131.2 113.0 124.7 107.2 128.1 118.8 116.6 123.0 124.0 126.6 126.4
geometries. For the structures with lower symmetry, we present mean values averaged over the two molecules in the irreducible part of the respective unit cell. Within the irreducible part of these crystals, variations exceeding 0.001 Å or 0.1° are only observed for very few quantities. Tables 2 and 3 both have only one joint column for the CS and C2 configuration since there is no difference in the plotted values, neither for the bond angles nor for the bond lengths. For comparison with the gas-phase molecule, Tables 2 and 3 also contain the bond lengths and bond angles, respectively, of gaseous guanine. In general, the similar results for the three crystalline phases indicate that the relaxation of the guanine molecules in the crystal is almost the same. From Tables 2 and 3, we conclude that the guanine molecules in the low-symmetry structures experience only tiny changes compared with those in the C2h structure. In comparison to the experimental data, we measure standard deviations of 1.4° and 4° for bond angles of heavy atoms and for bond angles of hydrogen atoms, respectively. The larger spread in the values for the hydrogen atoms is reasonable, since their uncertainty in the experimental data is considerably larger. For the heavy atoms, the reported estimated standard deviation is 1-1.5°. The bond lengths between heavy atoms obtained from experimental data and LDA calculations deviate by 0.018 Å which compares to the estimated error of 0.0100.015 Å in the experiments. The mean difference of the bond lengths of the hydrogen atoms has a larger value of 0.05 Å which might only partially be attributed to experimental uncertainty. From the computational point of view, however, this overestimation by 0.05 Å is also characteristic of the local approach to XC in LDA and can partially be overcome by including gradient corrections. However, a detailed comparison of various XC functionals is not the aim of this study. We turn to the comparison of gas-phase guanine and the molecule in the monohydrate crystal. First of all, we find that albeit bond lengths and bond angles are modified upon condensation, the guanine molecule remains intact. From Table 2, it becomes obvious that a XH‚‚‚A hydrogen bridge bond
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Figure 4. Vibrational density of states as described in the main text. Low-frequency range 0-1800 cm-1 (upper panel) and high-frequency range 1800-3600 cm-1 (lower panel). Results for the three different water column structures are plotted in different colors.
effects the XH bond length. The XH bond lengthening is the well-known attraction of the proton by electron lone pairs at the acceptor site. Only, H4 is not directly related to an acceptor, giving very similar results for the C8-H4 bond length in the gas phase and in the crystal. The hydrogen bond lengths H‚‚‚A of the other atoms cover the range of 1.62 to 1.79 Å in the high-symmetry crystal. Crystallization further leads to a shortening of the N1-C6 bond and an elongation of the C6-O6 bond as the main effects. This is explained by the hydrogen bond between the water columns and O6. Related thereto is the change in the bond angles N1-C6-C5 and C5-C6-O6. Even stronger effects are observed for the bond angles of hydrogen atoms which holds in particular for the amino group. In the crystalline phase, we observe a weaker pyramidalization of the amino group than in the gas phase. This effect mirrors the hydrogen bonding in the layers. The modifications of the molecular geometry of guanine upon crystallization by far exceed the variations among the three crystal structures. As a result, the structural influence of the water columns on the guanine molecules in the three different crystals is very similar as shown by the above comparison of the intramolecular bonding parameters and the lattice geometry. However, another important quantity is the total energy for the three structures. Here, we find that, from the energetic point of view, the most symmetric crystal is the least stable one. A lower energy is obtained in both low-symmetry structures. It costs approximately 0.15 eV per unit cell or 0.84 kcal per mole water molecules to move the hydrogens into the mid-oxygen position in the C2h structure. Since this holds for both low-symmetry structures, the potential for these hydrogens in-between the oxygens can be characterized as a double-well barrier. B. Dynamic Properties. A comprehensive overview on the dynamic properties can be gained from Figure 4. It shows the vibrational density of states (DOS) plotted for the complete frequency range. Below 1800 cm-1, there is practically no difference in the two graphs from the low-symmetry structures. A somewhat different DOS is obtained for the C2h crystal. Although it mainly follows the DOS of the other crystals rather
closely, there are some regions where they differ. These modifications can be traced back to vibrations including movements in the water columns. In the high-frequency part of Figure 4, this effect gets even more pronounced. The OH stretching modes around 2750 cm-1 are completely missing in the DOS of the high-symmetry C2h crystal. On the other hand, this crystal exhibits an artificial peak at 1940 cm-1 coming from an OH stretching not present in water molecules. We call this peak artificial because we do not expect such a peak in an experimental spectrum. A careful inspection of the vibrational problem shows that the mid-oxygen structure (i.e., the highsymmetry C2h crystal) is unstable because two frequencies turn out to be imaginary. The respective eigenvectors show that the hydrogen atoms are moved out of their position between the oxygens. This corresponds to our results from the previous section that lower total energies are found for the low-symmetry structures. The C2h geometry represents a saddle point on the total energy surface, and the forces on the displaced hydrogens obey a double well potential. In the calculations, the symmetry constraint prevents this structure from relaxing away from the saddle point in the total energy minimization. In contrast, the phonons associated with intramolecular vibrations of the guanine molecules are generally much less affected by the atomic structure of the H2O columns. This is documented in Supporting Information and will not be discussed further. Here, we concentrate on the discussion of the lowfrequency vibrations as displayed in Table 4, mainly for two reasons: First, they basically describe intermolecular vibrations and are therefore more sensitive to the actual bonding between the molecules. Second, because of their low frequencies, these modes can be occupied at room temperature and, hence, influence thermal properties and charge-transport characteristics of the crystal. The intermolecular vibrations can also indicate possible changes in the crystal bonding that can be detected immediately from changes in the mode frequency or the motion pattern of these modes. For the three symmetries of a guanine monohydrate crystal as discussed throughout this paper, the presentation of the lowest
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TABLE 4: Lowest Phonon Modes for the Guanine Monohydrate Crystal with Frequencies (in cm-1), Symmetry Classification, and Mode Descriptions: For Abbreviations, See Main Text C2 mode description Tb Ta Rc, Ta Butterfly
Tb Rc RL, Ta Rc; H2Ocol: Tb Rc RL, Ta RL H2Ocol: Tb/a; Tb H2Ocol: Tb/a RL; H2Ocol: Ta/b/c RL RS; H2Ocol: Tc Tc RS H2Ocol: Tc; RS H2Ocol: Tc Tb; H2Ocol: Tc Tb H2Ocol: Tc; Tc, RS H2Ocol: Tc; RS, Tc H2Ocol: Tc; Tc Tc RS; H2Ocol: Tc Tc, RS RS
CS
C2h
frequency
symmetry
frequency
symmetry
frequency
symmetry
187.2 184.6 179.6 176.2 170.5 168.1 161.2 170.0 156.5 152.2 148.4 142.9 137.6 125.0 122.1 120.4 115.9 107.2 103.6 99.2 84.9 85.1 74.6
B A A A B B A B A B B A B B A A B A B B A B B
187.0 184.6 179.7 176.2 170.3 168.8 160.9 168.5 156.5 152.1 149.1 142.6 137.5 125.1 122.1 119.4 115.4 107.3 103.8 98.6 84.9 85.0 75.0
A′′ A′ A′′ A′ A′ A′′ A′′ A′ A′ A′ A′′ A′′ A′′ A′′ A′ A′′ A′ A′′ A′ A′ A′ A′′ A′
187.7 186.5 180.3 176.4 169.1 168.7 160.5 168.2 156.7 152.2 148.6 143.8 138.0 124.6 122.4 116.3 115.2 106.8 103.4 94.7 85.6 85.1
Bg Ag Au Ag Bu Bg Au Bu Ag Bu Bg Au Bg Bg Ag Au Bu Au Bu Bu Ag Bg
73.3
Bu
73.0
A
69.6
A′ 71.0
Ag
69.9
A′′
57.8 56.2 48.7 45.3
A′′ A′′ A′′ A′
67.2 58.0 57.0 48.2 43.9
Au Bg Au Au Ag
68.8
A
57.7 57.7 49.7 46.0
B A A A
energy vibrations in Table 4 includes the description of the modes, their frequencies, and a classification with respect to symmetry operations. A more comprehensive listing of all modes is given in Supporting Information. The low-frequency part, in general, is dominated by intermolecular modes where the molecules move as a whole. The large effective masses related to these translations and/or rotations of the molecules give rise to small frequencies. For similar reasons the lowest intramolecular vibration in the crystal, more precisely the bending motion of the guanine molecule as a whole (the socalled butterfly mode), appears in this region as well. For the mode description of intermolecular modes, we use the term T (R) for translation (rotation) of the guanine molecules if not indicated differently as, for example, “H2O:T”, which denotes a translation of the water molecules or “H2Ocol:T”, which indicates that the water columns translate as a whole. The indices a, b, and c refer to the displacement direction more or less parallel to the crystal axes. For rotations, the index L (S) denotes the long (short) axis of the guanine molecule as the rotation axis within the guanine plane while c denotes the normal axis to the plane. Finally, due to the symmetry classification of the modes with C2h symmetry, one can identify IR-active and Raman-active modes as ungerade (u) and gerade (g) modes, respectively. The results in Table 4 show that for the majority of the modes one can observe very similar mode patterns in all three crystals. These modes are put together under the same mode description. Furthermore, the DOS below 250 cm-1 is rather similar for the three crystals. Hence, it follows that the influence of the water column arrangement on low-frequency modes is also small and that it is essentially restricted to modes with contributions from
the water. Further evidence is found in the respective frequencies where deviations among the structures rarely exceed a few cm-1. In conclusion, similar to the structural properties discussed in section III.A, the vibrational properties of guanine crystals are also only minimally influenced by the exact arrangement of the water columns. Even the energetically unfavorable C2h structure gives nearly the same results as the more stable CS and C2 crystals, especially in the low-energy region relevant for thermal and/or charge transport. C. Electronic Properties. The electronic DOS is plotted in Figure 5 for the three geometries of the H2O columns. Apparently, the graphs are again very similar among the structures. Once more, this indicates the localized nature and small magnitude of effects caused by structural modifications of the water columns. A closer inspection of the graphs reveals that the low-symmetry structures and the C2h crystal differ in their electronic DOS primarily in the lower energy region around -20 eV. These modifications are attributed to states centered at the water columns. In particular, we observe a broader band directly below -18.5 eV for the high-symmetry crystal. This band can be ascribed to the lowest-lying interacting H2O states. In the low-symmetry structures, these σ states couple much weaker along the columns resulting in a smaller bandwidth. Across a symmetric bridge, the overlap between these states is larger, and consequently, the coupling is enhanced. However, because of their energetic position, these states do not influence the highest occupied crystal orbital (HOCO) and the lowest unoccupied crystal orbital (LUCO) which are the most important orbitals for charge transport. Decomposing the DOS with respect to the molecules (not shown as a plot) further confirms that HOCO and LUCO are only derived from guanine orbitals. As
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Figure 5. Electronic density of states. The valence band maximum is set to zero.
Figure 6. Bandstructure of the guanine crystal in the C2h structure. The valence band maximum is set to zero. Inset: Irreducible part of the Brillouin zone with half reciprocal lattice vectors R′, b′, and c′.
an important consequence, it follows that for the three structures discussed throughout this paper the electronic DOS in the vicinity of the fundamental gap is nearly identical, as can be seen from Figure 5. In order to investigate the electronic properties further, we also calculate the crystal band structure. A plot of the gap region is displayed in Figure 6 for the C2h crystal. The definition of critical points in the Brillouin zone is given in the inset of Figure 6. At first, in agreement with the findings for the DOS, we state that the variation among the three configurations in the regions of the highest valence bands and lowest conduction bands is only on the order of 10 meV, hence they would be hardly distinguishable in Figure 6. Consequently, the figure shows only the band structure of the C2h crystal. Moreover, this accordance holds for the entire HOMO band as well as for the LUMO band each of which consists of four bands lying energetically close to each other. From these findings, it follows that the respective states in the C2 structure and the CS structure nearly coincide with their counterparts in the C2h crystal. Having discussed the negligible influence of the water molecules, we now turn our attention to the interaction between the guanine molecules themselves. There are two types of such interactions. First, we can expect an interaction within each guanine layer due to hydrogen bonding between the four guanine molecules in the unit cell. A good measure for their type of interaction is the Davydov splitting of the HOMO and LUMO bands at Γ. The energy difference between HOCO and HOCO-3 is found to be 0.16 eV, and the splitting between the LUCO
and LUCO+3 amounts to 0.11 eV. Even though these splittings are not small, compared with the overall dispersion of the HOMO and the LUMO bands, they are of minor importance. The strong band dispersions are a consequence of the other type of interaction, namely, the interaction between the guanine layers. In fact, the largest dispersion of the guanine states is observed in the Γ Z direction, that is, approximately the direction of the H2O columns. The reason for this is the strong coupling of the molecular states in that direction caused by the shape and the extent of the π orbitals. In that respect, the crystals are similar to the DNA itself. There, the interaction of the π orbitals along the helix axis has already been suggested to be a prerequisite for a one-dimensional conducting channel by Eley and Spivey.44 Moreover, doublets and triplets of guanines act as an effective hole trap due to their higher oxidation potential compared to single guanines,45 which results from the coupling of the molecular states. In the language of tight-binding theory, the interaction of the orbitals and the resulting band dispersion is quantified in terms of transfer integrals which can only be large for sufficiently overlapping wavefunctions. This overlap is maximal in the guanine stacking direction resulting in the strong band dispersion in Γ Z direction. Thereby, the bandwidth assigned to the HOMO band (0.83 eV) is significantly larger than the bandwidth of the LUMO band (0.38 eV). In order to illustrate what efficient coupling means we refer to three different situations where the bandwidths associated with guanine HOMOs have been calculated in the literature.
Guanine Crystals The first example is a part of a DNA double strand with eleven guanine molecules in a poly(G)-poly(C) chain. There the coupling was much less effective because of the geometry in the DNA resulting in a bandwidth of only 0.04 eV.5,46 The second situation is that of a guanine quadruple helix (G4 wire) where the guanine molecules of adjacent planes are rotated by 30°.17 The bandwidth resulting from 12 guanine HOMOs amounts to 0.26 eV there. The third example refers to a model situation where the guanine molecules of a one-dimensional stack are artificially positioned exactly on top of each other.21 Even though these cofacially aligned guanine molecules may not be stable, the large calculated bandwidth of approximately 1.1 eV shows how strong band dispersion emerges from an efficient coupling of π stacked guanine molecules. For the guanine monohydrate crystal considered in the present paper, the bandwidth of the highest valence bands comes close to this range. In fact, the HOMO bandwidth of about 830 meV is larger than those found in comparable calculations for oligoacene crystals.37 This indicates that such guanine crystals should indeed be considered as organic crystals with potential electronic applications, at least with hole transport in the stacking direction. In the a′b′ plane, the energy dispersion is approximately 1 order of magnitude smaller. In the language of tight-binding theory, the respective transfer integrals are smaller. Comparing the magnitude of the transfer integrals in different directions, we can estimate from a transport theory43,47 that the resulting band-like transport in terms of the conductivity in the a′b′ plane is roughly 2 orders of magnitude worse than perpendicular thereto. Besides transport properties, many other characteristics, for example, optical properties, are determined by the fundamental band gap of the crystal. From Figure 6, one obtains a direct gap at Γ, but we note that the interband energies for k within the a′b′ plane possess similar values. The direct KS gap amounts to 2.73 eV. This is the energy difference between KS eigenvalues but not the true quasiparticle or optical gap as outlined in section II.3. According to our computations, for the gas-phase molecule, the KS gap amounts to 3.90 eV, thus, 1.17 eV larger as in the crystal. In the ∆SCF approach, the vertical gap energy is determined to 7.72 eV. The quasiparticle correction for the gas-phase molecule is therefore calculated to 3.82 eV. If we assume that this quantity is almost unchanged in the crystal, we can add the quasiparticle correction for the molecule to the KS gap of crystal as well. The resulting transport or quasiparticle gap then amounts to 6.55 eV. This value may be regarded as an upper limit as the increased screening in the crystal may give rise to a small reduction of the gap energy. IV. Summary In this paper, we have studied guanine in the solid state condensed in guanine monohydrate crystals. In particular, we have presented theoretical results from DFT calculations for the static and dynamic properties as well as the electronic DOS and band structure of these crystals. Throughout the paper, we have put a special focus on the comparison between different crystal geometries caused by different arrangements of the water columns within the crystals. For the structural properties of guanine crystals, we have presented LDA and GGA results for the lattice parameters. From a comparison to the available experimental data, we conclude that the intermolecular interactions are reliably described within LDA which underestimates the experimental lattice constants by 3-5%. Similarly, also the intramolecular geometries calculated within LDA are found to be in good overall agreement
J. Phys. Chem. B, Vol. 112, No. 5, 2008 1547 with the experiment with deviations of up to only 0.018 Å and 1.4° for the bond lengths and bond angles of the heavy atoms, respectively. From a detailed comparison of the covalent bonds in the three crystal structures (C2h, CS, C2) with those in gasphase guanine, we conclude that the modifications of the molecular geometries induced by the crystallization by far exceed the variations among the various crystal structures. We have given a comprehensive overview of the vibrational properties of guanine crystals including predictions for the frequencies of all intra- and intermolecular vibrations. The influence of the shape of the water columns has been found to be of minor importance. Its impact on the vibrations is strong for some of the modes with contributing water vibrations but very weak for the majority of the modes, in particular, the lowenergy intermolecular phonons relevant for thermal and/or charge transport. We have pointed out that the symmetric mid-oxygen configuration of the guanine crystal is not stable. A real crystal might consist of more or less big domains of the less symmetric CS and C2 types separated by water vacancies. Considering the smeared water oxygen positions and the unresolved water hydrogens in the experiments, this might be a possible scenario for a real crystal structure where the guanine sublattice exhibit more or less the full C2h symmetry. We have observed an influence of the water molecules on the geometry change upon condensation from the gas phase, but the direct influence of the water positions in the columns on structural, dynamic, and electronic properties is negligible. In our studies of the electronic structure, we have paid special attention to states in the vicinity of the fundamental band gap because of their importance for the charge carrier transport through guanine crystals. We found that the guanine HOMO (LUMO) transforms upon condensation into a HOMO band (LUMO band) accompanied with large bandwidths, in particular, for the HOMO (830 meV). The strong energy dispersion for the hole states and a moderate dispersion for the electron states in the stacking direction are a direct consequence of the interaction of the molecular states and have been discussed. Along the other crystallographic directions, only weak energy dispersion occurs implying that charge transport in guanine crystals will mainly occur between the various guanine layers but hardly within each of the layers. In conclusion, our results suggest that significant charge transport in van der Waals bonded organic molecular crystals should be possible not only for herringbone-stacked materials (such as oligoacene crystals) but also in layered crystals (such as DNA base crystals) along the stacking direction of the molecules. Acknowledgment. We would like to thank the Deutsche Forschungsgemeinschaft for financial support (Project No. HA 2900/3-2). This work was supported by the European Community within the framework of the Network of Excellence NANOQUANTA (Contract No. NMP4-CT-2004-500198). Grants for computer time from the Leibniz-Rechenzentrum Mu¨nchen are gratefully acknowledged. Supporting Information Available: Table of frequencies and mode descriptions for the three guanine crystal structures. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Mirkin, C. A.; Letsinger, R. L.; Mucic, R. C.; Storhorff, J. J. Nature (London) 1996, 382, 607.
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