ORCA labs 10 weak interactions...

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]:


“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:


EI= E

S+ E

PL+ E

X+ E

CT+ E

DISP+ E

MIX (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).

EPL 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.. EPL

decreases with –r-‐4.

EX 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. EX

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.

EDISP 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. EDISP is isotropic and

decreases with –r-‐6.

EMIX 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


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


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)!

1r12"

1r 6 (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 EI of a complex AB which is build from the monomers A

and B is defined as:

EI= E(R

AB)!E(R

A)!E(R

B) (3)

E R

AB( ) is the total energy of the complex AB, E R

A( ) and E R

B( ) 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


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(RÁ,a), the energy of the isolated monomer A with the geometry

RÁ it has in the complex AB. For this calculation only the basis functions a for

molecule A are used.

2.) One calculates E(RÁ,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(RÁ,a) -‐ E(RÁ,ab)} + {E(R´B,b) -‐ E(R´B,ab)} (4)

And for the counterpoise corrected interaction energy EIcc we get:

EIcc = EI + EBSSE (5)

The calculation of E(RÁa) 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(RÁ,ab) and E(R´B,ab) can be more


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):

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.

! RKS B3LYP TZVP TightSCF * xyz 0 1 O 0.081307 -0.360387 0.357004 H -0.259589 0.505266 0.499968 H 0.926815 -0.246293 -0.052696 O: 2.753726 -0.032808 -0.943497 H: 3.521598 -0.305170 -0.469305 H: 2.859663 -0.336280 -1.829698 end


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.


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


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


!

12= ! P

2!P

1( )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

P

i= !!!14!"#

(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-"4!"!"

in

!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

!="!14!"

En

(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


● Acetone

● n-‐Octane

● Benzene

● OH-‐

● Cl-‐

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:


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

! RKS BP RI TZVPP TZVPP/C COSMP(water) TightSCF %tddft Nroots 5 Maxdim 50 End *xyz 0 1 . . *


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?

ORCA labs 10 weak interactions...

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