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Intermolecular and Weak Interactions 1 1 Computer Experiment 10: Intermolecular and Weak Interactions 1.1 Background In this section we treat intermolecular and weak interactions, this means non-­‐covalent and non ionic interactions between different atoms or molecules. We will focus our attention to the two most familiar representatives of such interactions: Hydrogen bonds and van der Waals or dispersion interactions. Other weak intermolecular interactions like π-­‐π stacking or dipole-­‐dipole interactions are closely related to these two basis types of interaction. 1.1.1
Hydrogen bonds Hydrogen bonds belong to the most important non-­‐covalent intermolecular interactions. They are operative in determining molecular conformation, molecular aggregation, and the function of a vast number of chemical systems ranging from inorganic to biological. In the majority of cases hydrogen bonding occurs between a polar bond Xδ -­‐-­‐Hδ + and an electron rich region Bδ -­‐. Hydrogen bonding can be understood as a donor-­‐acceptor interaction. Therefore we will denote X-­‐H as the (hydrogen) donor and B as the (hydrogen) acceptor. Definition: The first definition of the hydrogen bond was given by Pauling [1]: “Under certain conditions a hydrogen atom is attracted by rather strong forces to two atoms, instead of only one, so that it may be considered to be acting as a bond between them.” He further added that “it is now recognized that the hydrogen atom, with only one stable orbital (the 1s orbital), can form only one covalent bond, that the hydrogen bond is largely ionic in character, and that it is formed only between the most electronegative atoms.” This definition implies that the hydrogen bond consists of two components: A hydrogen bond donor X-­‐H, where the H atom is covalently bonded to an atom X with a large electronegativity, and another electron rich atom B, the hydrogen bond acceptor. Typical examples are hydrogen bonds between water molecules or between amide groups in peptides. However, the second statement by Pauling limits the hydrogen bond to a few atoms with high electronegativity. Because nowadays many other bonding situations such as C-­‐H…B are also referred to as hydrogen bond further definitions were suggested. The most recent originates from a IUPAC conference in the year 2005 [2]: Intermolecular and Weak Interactions 2 “Hydrogen bonding occurs when an electron deficient hydrogen that is bonded to an atom, has an attractive interaction with another electron rich region either within the same or another molecular entity” Hydrogen bonding can occur in many different fashions. Mostly one distinguishes between strong, moderate and weak hydrogen bonds. Moderate or normal hydrogen bonds have interaction energies between 20 and 60 kJ/mol, common examples are hydrogen bonds between water molecules or amide groups. Weak hydrogen bonds have interaction energies below 20 kJ/mol. An example for a weak hydrogen is the interaction between a methyl group (C-­‐H as acceptor) and the π-­‐electrons of benzene. Strong hydrogen bonds are characterized by the fact that the H atom lies close to the mid point of the A…B line of centers (an example is the negatively charged complex [F…H…F]-­‐). The interaction energy is larger than 60 kJ/mol. A further important point concerning to hydrogen bonds is their cooperativity. The properties of a network of n hydrogen bonds are not the sum of n isolated hydrogen bonds. This cooperativity or nonadditivity is caused by the polarizability or charge transfer character of hydrogen bonding. One example for this cooperativity is the following hydrogen bonded chain: Yδ -­‐-­‐Hδ +…Xδ -­‐-­‐Hδ +…Bδ -­‐ Both Yδ -­‐-­‐Hδ -­‐ and Bδ -­‐ effect a polarization of the middle Xδ -­‐-­‐Hδ + group. Through this opposite polarization both hydrogen bonds became stronger. Theory of hydrogen bonds Hydrogen bonding is a quite complex interaction. In earlier days it was believed that hydrogen bonding is a purely electrostatic interaction between Xδ -­‐-­‐Hδ + and Bδ -­‐ (In fact electrostatic models are often able to predict reasonable geometries for hydrogen bonds). However a pure electrostatic model is not able to explain e.g. the spectroscopic properties of hydrogen bonds. A charge transfer from Bδ -­‐ to Xδ -­‐-­‐Hδ + is certainly also involved in hydrogen bonding (This is the reason why some authors call Bδ -­‐ the (electron) donor and Xδ -­‐-­‐Hδ + the (electron) acceptor). Furthermore, of course it is not possible to treat any intermolecular interactions without dispersion and repulsion forces. Hence hydrogen bonding is the sum of many different effects. Morokuma, in 1977, introduced an energy decomposition scheme [3] which divides the interaction energy EI in the following components: Intermolecular and Weak Interactions E I = ES + E PL + E X + ECT + E DISP + E MIX 3 (1) Morokuma explains the physical meanings of these components as: ES is the electrostatic interaction, i.e. the interaction between the undistorted electron distribution of the donor (X-­‐H) and that of the acceptor (B). This contribution includes the interaction of all permanent charges and multipoles, such as dipole-­‐dipole, dipole-­‐
quadrupole, etc.. ES is oriented and has a long range (it decreases with -­‐r-­‐3 for dipole-­‐
dipole and with -­‐r-­‐2 for dipole–monopole interactions). E PL is the polarization interaction, i.e. the effect of the distortion (polarization) of the electron distribution of the donor by the acceptor, the distortion of the acceptor by the donor, and the higher order coupling resulting from such distortions. This component includes the interactions between all permanent charges or multipoles and induced multipoles, such as dipole-­‐induced dipole, quadrupole-­‐induced dipole, etc.. E PL decreases with –r-­‐4. E X is the “exchange repulsion”, i.e. the interaction caused by exchange of electrons between the acceptor and the donor. More physically, this is the short-­‐range repulsion due to overlap of electron distribution of the donor with that of the acceptor. E X increases with +r12. ECT is the charge transfer or electron delocalization interaction, i.e. the interaction caused by charge transfer from occupied orbitals of the acceptor to vacant orbitals of the donor, and from occupied orbitals of the donor to vacant orbitals of the acceptor, and the higher coupled interactions. The ECT contribution decreases approximately with -­‐e-­‐r. E DISP is the dispersion energy, the stabilization due to the correlation of electronic motions in the acceptor and the donor. This interaction is described by simultaneous and correlated excitations of electrons in both molecules. E DISP is isotropic and decreases with –r-­‐6. E MIX includes higher order interactions between various components. One must keep in mind that each hydrogen bond has its own weighting of these components (depending on the acceptor, the donor and their environment (geometry). In particular the different distance dependencies of the components are important. For Intermolecular and Weak Interactions 4 example a hydrogen bond gets more electrostatic if the distance r between the donor and the acceptor is increased. 1.1.2
Van der Waals interaction In Chemistry, the term van der Waals interaction originally referred to all intermolecular interactions. According to this one definition a van der Waals interaction is: “All intermolecular interactions and interaction between two atoms/groups in a molecule that are not directly bonded are called van der Waals interactions.” [2] However, the term van der Waals interaction is often only used for intermolecular interactions which arise from the polarization of molecules into dipoles or multipoles. In this case one also speaks of dispersion or London interaction: “Interaction between two atoms/groups arising primarily due to dispersive forces is called London interaction.” [2] In this script we will use the term van der Waals interaction as synonum for London or dispersion interaction. Even though dispersion interactions are the weakest non-­‐
covalent interactions they are very important in chemistry, since they are the only attractive force at large distance present between neutral atoms like noble gases or nonpolar molecules like alkanes. For example without dispersion interaction it would be no possibility to obtain noble gases in a liquid form. Furthermore dispersion forces become stronger as the atoms or molecules become larger. This is due to the increased polarizability of molecules with larger, more dispersed electron clouds. Theory of the Dispersion interaction Dispersion or London force is an attractive interaction which arises from an interplay between electrons belonging to the densities of two otherwise non-­‐interacting atoms or molecules. It originates from the fact that electron density is even in nonpolar molecules not evenly distributed in space. At intermediate distances the motion of electrons in one molecule induces slight perturbations in the otherwise evenly distributed electron densities of the neighbouring molecule. This correlation leads to a temporary dipole moment. The induced dipole, in turn, induces a polarization of the electron density of the other molecule. Through this it comes to an attractive interaction between the two molecules. As one knows from Physical Chemistry such induced dipole-­‐induced dipole interactions decay with –r-­‐6 (r is the distance between the two molecules). For completeness one Intermolecular and Weak Interactions 5 should mention that the presence of interactions from higher order electric moments like induced quadrupole-­‐induced dipole or induced quadrupole-­‐induced quadrupole interactions leads also to terms which vary with –r-­‐8, –r-­‐10 and so on. The repulsion interaction between the electrons of the two interacting molecules acts in the opposite direction. The Pauli repulsion or exchange interaction arises with +r12. Dispersion and exchange interaction are the sole electronic interactions which occur in all molecules. Together they give the well known Lennard-­‐Jones Potential V(r) !
1
1
" 6 12
r
r
(2) where higher order dispersion interactions are neglected. Calculation of interaction energies: The most common Ansatz for the calculation of interaction energies of hydrogen or van der Waals bonded complexes is the supermolecular approach. In this Ansatz the total intrinsic interaction energy E I of a complex AB which is build from the monomers A and B is defined as: E I = E(RAB )! E(RA )! E(RB ) (3) E (RAB ) is the total energy of the complex AB, E (RA ) and E (RB ) are the total energies of the isolated molecules A and B. It should be noticed that the supermolecular Ansatz describes the total interaction energy of the complex. For example it is not possible to determine individual hydrogen bond energies if there is more than one hydrogen bond broken upon dissociation. Moreover the supermolecular Ansatz can not be used for the calculation of intramolecular interactions. BSSE and Counterpoise Correction: Calculation of the interaction energy by means of the supermolecular Ansatz involves an important inconsistency. When the total energy for the isolated monomers A or B is calculated only the basis functions of the particular molecule (A or B) are available to describe each electronic spatial orbital. Whereas when the total energy of the complex AB is calculated the basis functions of one molecule can help compensate for the basis set incompleteness on the other molecule, and vice versa. In effect the basis set of the complex is larger than that one of each monomer. This produces an artificial lowering of the complex energy relative to that of the separated monomers and as a consequence thereof an overestimation of the interaction energy. This effect is called basis-­‐set superposition error (BSSE). It should be clear that the larger the used basis set the Intermolecular and Weak Interactions 6 smaller is the BSSE. In the limit of a complete basis set the BSSE would vanish since adding basis functions would not lead to an improvement in this case. However, calculations with a (nearly) complete basis set are not possible (or not reasonable) in the majority of cases. In order to get accurate values for the interaction energies it is necessary to find a practicable correction for the BSSE. The most common used method for an approximated determination of the BSSE is the Counterpoise correction [4]. In the counterpoise correction one estimates the above described artificial lowering of the total energy of the complex AB. For this purpose four additional calculations are necessary: 1.) One calculates E(R´A,a), the energy of the isolated monomer A with the geometry R´A it has in the complex AB. For this calculation only the basis functions a for molecule A are used. 2.) One calculates E(R´A,ab), the energy of the isolated monomer A with the geometry it has in the complex AB. However, now the complete basis ab of the complex AB, that means the basis functions for molecule A and B are used. The basis functions b of molecule B are located at the corresponding positions of the nuclei, but the B nuclei are not present. These positions in space where only basis functions but no nuclei are located are often called ghost atoms or ghost orbitals. 3.) According to step 1.) one calculates E(R´B,b), the energy of B with the geometry it has in the complex AB (of course with the basis functions b). 4.) According to step 2.) one calculates the energy E(R´B,ab) of B in the complete basis ab of AB. For the BSSE we get: EBSSE = {E(R´A,a) -­‐ E(R´A,ab)} + {E(R´B,b) -­‐ E(R´B,ab)} (4) (5) And for the counterpoise corrected interaction energy EIcc we get: EIcc = EI + EBSSE The calculation of E(R´Aa) and E(R´B,b) is quite easy. One deletes the coordinates of A respectively B from the optimized structure AB and performs a normal calculation with the chosen basis set. The calculation of E(R´A,ab) and E(R´B,ab) can be more Intermolecular and Weak Interactions 7 complicated because the ghost atoms must be defined. ORCA offers a reasonably convenient solution: If one wants a ghost atom with specific basis functions at certain coordinates it is only necessary to add in the input file a colon “ : “ behind the atom symbol. Example input for calculation of E(R´A,ab) (AB is a water dimmer): ! RKS B3LYP TZVP TightSCF
* xyz 0 1
O
0.081307 -0.360387
H
-0.259589 0.505266
H
0.926815 -0.246293
O:
2.753726 -0.032808
H:
3.521598 -0.305170
H:
2.859663 -0.336280
end
0.357004
0.499968
-0.052696
-0.943497
-0.469305
-1.829698
Computational Methods: Before one starts with the calculation of hydrogen bonded or van der Waals bonded systems one must choose a suitable computational method. Nowadays density functional theory is the most common used method for standard computational calculations. However, DFT methods have some shortcomings which narrow their ability to describe such situations. The main disadvantage of DFT methods in this context is their deficient description of dispersion. Since the van der Waals interaction is a pure dispersion effect it should be obvious that DFT is the wrong choice for the calculation of such interactions. One must mention that one can get quite good results for von der Waals complexes with some functionals. However, it is not clear, if this is “the right result for the right reason”. The situation is more complicated for hydrogen bonds since their interaction energy can be described as the sum of different contributions. Depending on the situation DFT methods are more or less suitable: For weak hydrogen bonds, where dispersion interactions are more important, DFT is a bad choice, whereas it delivers good results for normal hydrogen bonds which are a mainly electrostatic interaction. However, even in these cases one should verify DFT results with a method with a better description of electron correlation. Coupled Cluster methods like CCSD(T) with a large basis set would be an excellent choice. However, due to its huge computational costs such calculations are not practicable in most cases. Instead of one common uses MP2 calculations for the calculation of van der Waals and hydrogen bonded complexes. MP2 combines a good description of correlation effects like dispersion with moderate computational costs. Intermolecular and Weak Interactions 8 Further Reading: The physical background of the underlying forces of weak interactions is well explained in many Physical Chemistry textbooks. A more detailed description of hydrogen bonding can be obtained from an excellent review article from Thomas Steiner [5] or some textbooks [6-­‐9]. Accurate computation of weak interactions is still a challenge in Quantum Chemistry. For the problems of DFT with the description of weak interactions we especially refer to the book by Koch and Holthausen [10]. 1.2 Description of the Experiment 1.2.1
Exercise 1: Investigation of the water dimer In the first exercise we will study the water dimer. Since it is well known that different conformations are possible we will begin the calculation at different starting points. Proceeding: •
Build a z-­‐matrix for different conformations of the water dimer. Try to create at least one linear, one cyclic and one bifurcated dimer. 290 pm is a reasonable start value for the distance ROO between the two oxygen. •
Perform geometry optimizations of the different conformers. Use the B3LYP, RHF, MP2 and CCSD(T) methods1. Update the basis set from SVP to TZVP to TZVPP to aug-­‐
TZVPP. Study how the BSSE behaves with basis sets of increasing size. Do the diffuse functions in aug-­‐TZVPP improve the results? •
Calculate the interaction energy by means of the supermolecular approach. •
Calculate the counterpoise corrected interaction energies •
Compare the results (interaction energy, BSSE and geometry) which you have obtained with the different methods among each other and compare them to experimental result (interaction energy = 5.0±0.7 kcal/mol = 20.9±2.9 kJ/mol)2 1.2.2
Exercise 2: Noble gas dimers In this section we study the potential energy surface (PES) of the He and the Ne dimer. For this purpose we calculate the energy of the He and Ne dimer at different interatomic 1
Note that you obtain the RHF results as a byproduct of the MP2 calculation. 2
Quoted from Mas, E.M.; Bukowski, R.; Szalewicz, K.; Groenenboom, G.C.; Wormer, P.E.S.; van der Avoird, A. (2000), J. Chem. Phys., 113, 6687. Intermolecular and Weak Interactions 9 distances (RHe-­‐He or RNe-­‐Ne). The calculations of this exercise should be performed both with MP2 and B3LYP each with a TZVPP basis (for higher accuracy aug-­‐TZVPP). •
Begin the calculation with an interatomic distance (RHe-­‐He / RNe-­‐Ne) which is twice the van der Waals radius of the treated atoms. •
Vary (increase and decrease) this distance in steps of 10 pm. You should consider at least 10 points. •
Calculate for each point the interaction energy (EI and EICC). •
Nearby the minimum you can choose a closer meshed net for your calculation. •
Compare the results of the MP2, CCSD(T) and B3LYP calculation and discuss the importance of the BSSE in this case. Compare your results to accurate computational data (Basis-­‐set limit MP2 values:3 He-­‐dimer: 0.06 kJ/mol, R(He-­‐He)=3.061 Angström; Ne-­‐dimer: 0.23 kJ/mol, R(Ne-­‐Ne): 3.202 Angström; CCSD(T)-­‐values:4 He-­‐dimer: 0.09 kJ/mol, R(He-­‐He): 2.977 Angström; Ne-­‐dimer: 0.34 kJ/mol, R(Ne-­‐Ne)= 3.101 Angström). Literatur: [1] L. Pauling, The Nature of the Chemical Bond, Cornell University Press, Ithaca, NY, 1939 [2] E. Arunan, R. Klein, IUPAC workshop “Hydrogen Bonding and Other Molecular Interactions”, Pisa, 2005 [3] H. Umeyama, K. Morokuma, JACS, 1977, 99, 1316 [4] a) S. F. Boys, F. Bernardi, Mol. Phys. 1970, 19, 553 b) F. B. van Duijneveldt, J. G.M. van Duijneveldt-­‐van de Rijdt, J. H. van Lenthe, Chem. Rev. 1994, 94, 1873 [5] Th. Steiner, Angew. Chem. 2002, 114, 50 [6] G. A. Jeffrey, W. Saenger, Hydrogen Bonding in Biological Structures, Springer, Berlin, 1991 [7] G. A. Jeffrey, An Introduction to Hydrogen Bonding, Oxford University Press, Oxford, 1997 [8] S. Scheiner, Hydrogen Bonding. A Theoretical Perspective, Oxford University Press, Oxford, 1997 [9] G. R. Desirju, Th. Steiner, The Weak Hydrogen Bond, Oxford University Press, Oxford, 2001 [10] W. Koch, M. C. Holthausen, A Chemist’s Guide to Density Functional Theory, Wiley-­‐
VCH Weinheim, 2000 3
Tew, D.P.; Klopper, W. (2006) J. Chem. Phys., 125, 094302 4
Van Mourik, T.; Wilson, A.; Dunning, T.H.(1999) Molec. Phys., 96, 529 Intermolecular and Weak Interactions 2 Computer Experiment 11: Ions and Solvation Effects 2.1 Background 2.1.1
Introduction Solvent effects play an important role in all areas of chemistry. In it's broadest sense solvent effects can be defined as the change of a solute's properties in the presence of a solvent, compared to the gas phase. This can relate to changes in geometry, as well as in electric and magnetic properties. In the field of quantum chemistry people looked for ways to overcome the necessity of adding explicit solvent molecules, when trying to include solvent effects into their systems. These implicit solvent models usually treat the molecule's surroundings as a dielectric continuum, which reacts to the solute's charge distribution and in turn influences the solutes charge distribution. Two popular approaches of these dielectric models are the Polarizable Continuum Model (PCM), and Conductor-­‐like Screening Models (COSMO). In these models the solute's cavity is either represented by a series of atom-­‐centered spheres or by a solvent-­‐accessible / solvent-­‐excluding surface. The solute/solvent boundary is divided into small surface elements and the Poisson Equation is solved on each of them to yield an apparent surface charge (ASC). The COSMO method treats the surrounding continuum as conductor which greatly facilitates the solution of the electrostatic equations. The ASCs calculated this way are later modified to yield a result for a dielectric medium. These models can generally reproduce the experimental solvation energies for small charged molecules and are precise up to 1 kcal/mol for small uncharged molecules. 2.1.2
Theory The most versatile approach to implement a continuum solvent model are apparent surface charges (ASC). These can be derived by noting that the charge associated with each surface part, can be expressed as the difference between the polarization vectors at regions where different dielectric media come into contact:5 5
For an introduction into the treatment of dielectric media, see for example Feynman, “Lectures in Physics”, Vol. 2 10 Intermolecular and Weak Interactions 11 !12 = !(P2 ! P1 ) n12 (6) here σ is a charge distribution on the cavity surface, and n12 the normal vector from one region to the next. The polarization vector is related to the potential, which is generated by its region by Pi = !
! !1
"# 4!
(7) In this case, only two regions exist, namely !1 = 1; !2 = ! for the inside of the cavity and the outside, so the boundary charges can be calculated by !=
1-" !"in
4!" !n
(8) where !in is generated by the solute's charge distribution and the generated surface charges, or written in terms of the electric field !=
" !1
En 4!"
(9) These surface charges can then be used in quantum chemical calculations to influence the wavefunction which in turn gives rise to an new potential on the surface charges. This process is continued until self consistensy is reached (self-­‐
consistent reaction field, SCRF). In the framework of the COSMO model the surrounding is treated as a conductor, such that the potential vanishes at the surface. Due to this condition the equations become much easier to solve and also much easier to implement into quantum chemical codes. 2.2 Description of the Experiment 2.2.1
Calculation of Solvation Energies of Small Molecules Calculate the solvation energy of the following molecules: ● Water Intermolecular and Weak Interactions ●
Acetone ●
n-­‐Octane ●
Benzene ●
OH-­‐ ●
Cl-­‐ 12 The calculation includes several steps: 1. Optimization of the molecule's geometry in the gas phase using Density Functional Theory (B3LYP functional) with the SVP basis set: ! RKS B3LYP SVP TightSCF TightOpt
2. Optimization of the structure in solution using the COSMO solvation model at B3LYP/SVP level: ! RKS B3LYP SVP COSMO(water) TightSCF TightOpt
Using the final energies resulting from these calculations the electrostatic energy of solvation can be calculated. What do you observe if you look at the absolute numbers? Do the results for Benzene look odd to you? Which contributions do you think are missing to compute a Free Energy of solvation? 2.2.2
Solvent Shifts on Electronic Spectra Some molecules show a different absorption spectrum when immersed in liquids of different polarity. One of these molecules is One of the fastest ways to calculate a π → π* transition is the TD-­‐DFT method. An ORCA input for this calculation may be constructed as follows: Intermolecular and Weak Interactions 13 ! RKS BP RI TZVPP TZVPP/C COSMP(water) TightSCF
%tddft
Nroots 5
Maxdim 50
End
*xyz 0 1
.
.
*
The transitition that we are studying is essentially the HOMO-­‐LUMO transition and is the first excited state of the system. The solvatochromic shift is calculated as the energy difference between the excitation energy in hexane and water (just put in COSMO(hexane)). To produce reasonable results it is absolutely necessary to optimize the geometry of the molecule in the solvent under consideration. Thus, the geometric relaxation in a different solvents provides an important contribution to the observed solvent shifts.6 The experimental band maxima are observed at 20830 wavenumbers in hexane and 18830 wavenumbers in water. Do the calculation results differ from the experimental values? If so, how do you explain the discrepancies? 2.2.3
Calculation of Glycine in Gas-­‐ and Liquid Phase As one of the twenty naturally occurring amino acids glycine is a molecule of high biological significance. The aminoacids are composed -­‐ in their monomeric forms – of a central carbon atom to which an amino group, a carboxyl group, a hydrogen and a variable sidechain are attached. The peptide bond between these monomers is created as an amide condensation between the carboxyl group of one amino acid and the amino group of another one. The smallest amino-­‐acid is glycine, in which the side chain is only made up by an additional hydrogen atom. In its natural surrounding, that is -­‐ in solution -­‐ glycine takes the form of a 'zwitter' ion. This means, that both, the amino group and the carboxyl group are present in their ionic form, although the molecule as a whole remains neutral. 6
For experimental results see Reichardt, “Solvents and Solvent Effects in Organic Chemistry”, p. 335 Intermolecular and Weak Interactions 14 Figure 1: The neutral and zwitterionic forms of glycine. Thus this molecule presents a demanding test for an implicit solvent model, that is, can it stabilize the glycine molecule in its zwitterionic form and therefore predict the correct energetic preference? To answer this question, calculate (optimize) the structure of the glycine molecule in the gas phase and in solution with different dielectric constants. What is the critical value of the dielectric constant where the preference changes from the neutral to the zitterionic form?