Computational Module ITutorial day-3

Department of ChemistryUniversity of Zurich, Switzerland, 2018

What is thermochemistry?Frequency calculationsConformational analysis

Modeling chemistry in solution (PCM)

What is thermochemistry?

Thermochemistry is the study of energy changes involved in chemical reactions.Thermochemistry is used to predict whether a reaction is spontaneous ornon-spontaneous, favorable or unfavorable.

Thermodynamic Terms:• Chemical system: open (energy and matter can move in or out), closed (only

energy can move in or out), or isolated (neither energy or matter can move in orout)

• Reaction: exothermic (energy is released) or endothermic (energy is absorbed)• Heat Capacity (cal/K): the amount of heat needed to raise the temperature of one

kilogram of mass by one kelvin.• Enthalpy H (cal): a thermodynamic quantity equivalent to the total heat content of

a system. ∆H of a process is equivalent to its heat change at a constant pressure(H = U + P*V)

• Entropy S (cal/K): a thermodynamic function, which can be viewed as a measureof randomness or disorder, and describes the number of arrangements (positionand/or energy levels) that are available to a system existing in a given state.

• Gibbs free energy G (cal): it has no physical reality as a property of matter - thesign of ∆G indicates the direction of a chemical reaction and determines if areaction is spontaneous or not (G = H - T*S)

1 / 15

Examples

Figure: Exothermic reaction

Figure: Endothermic reaction

Figure: Exothermic process: rust

Figure: Endothermic process: cooking an egg

2 / 15

Example of Gaussian energy output

Figure: Energy values obtained in output from a Gaussian calculation

• The zero-point energy (ZPE) is a correction to the electronic energy of themolecule to account for the effects of molecular vibrations, that are present evenat 0 K. Final predicted energies must always include a scaled zero-point orthermal energy correction.

• The thermal correction to the Energy includes the effects of molecular translation,rotation and vibration at the specified temperature and pressure. It alreadyincludes the zero-point energy: E = Eelectr + ZPE + Evib + Erot + Etrans

3 / 15

Frequency calculations

• Frequency calculation can provide many info: IR/Raman spectra, force constants,stationary points, thermodynamic quantities (enthalpy, entropy, etc)

• A frequency calculation has to be performed on a previously optimised structure(use the same basis-set)

• By default, the analysis is carried at 298.15 K and 1 atmosphere of pressure,using the principal isotope for each element type

• Scaling factors are needed to eliminate known systematic errors in the calculatedfrequencies (the scaling factors are basis-set dependent)

Figure: Recommended scaling factors for frequencies and zero-point energies, for differentcalculation types

4 / 15

Stationary points I

• A stationary point can be characterised, in the output, from:• the number of imaginary frequencies• the normal mode corresponding to the imaginary frequency

• If any of the frequency values are less than zero, these frequencies are known asimaginary frequencies

• By definition, a structure which has n imaginary frequencies is an nth order saddlepoint. Thus, the minimum will have zero imaginary frequencies, and an ordinarytransition structure (TS) will have one imaginary frequency since it is a first ordersaddle point

• Be careful: the found TS may not be related to the reactants and products ofinterest. The nature of the saddle point can be understood looking at the normalmode corresponding to the imaginary frequency (animated vibrations; IRC)

5 / 15

Stationary points II

6 / 15

Conformational Analysis I - SCAN• SCAN: PES scans allow you to explore a region of the potential energy surface,

corresponding to the process in which you are interested. PES scans do notinclude a geometry optimisation

• Rigid SCAN: all coordinates are kept frozen, except for the particular coordinatebeing scanned. A single point energy calculation is performed for each generatedstructure

• Relaxed SCAN: the scan coordinates are kept frozen, while the others areoptimized. Each optimization locates the minimum energy geometry with thescanned parameters set to specific values.

• SCAN calculations provide insights into the structure of the PES, but they do notdefine the lowest energy path between two structures, that need to be obtainedfrom intrinsic reaction coordinate (IRC) calculations

7 / 15

Conformational Analysis: examples

Figure: Conformation analysis of Butane

8 / 15

Intrinsic Reaction Coordinate (IRC)

• More precise method to determine which points on a potential energy surface(PES) are connected by a certain transition structure (TS)

• An IRC calculation starts at the saddle point and follows the path in both directionsfrom the TS, optimizing the geometry of the system along the way. In this way twominima on the PES are surely connected by a path passing through the TS

• Be careful: two minima on the PES can have more than one reaction paths thatconnects them, with different TS through which the reaction evolves

• Once you have understood which minima the TS connects, you can go oncalculating the activation energy of the reaction, comparing the (zero-pointcorrected) energies of the reactants and of the TS

• In Gaussian, the following types of calculations have to be done, in order:optimize -> frequency -> IRC

9 / 15

Modelling systems in solution

• System: solution (a solute is a substance dissolved in another substance, knownas a solvent)

• As in classical electrostatics (P is the polarization function of the medium, ε is thepermittivity):

P =ε− 14π

E

• Electric polarization: slight relative shift of positive and negative electric charge inopposite directions within an insulator, or dielectric, induced by an external electricfield

10 / 15

Quantum Mechanical Continuum Solvation Models• Continuum model: model in which many degrees of freedom of the constituent

particles are described in a continuous way (usually with a distribution function)• Focused model: focused part + remainder -> there is no need to get a detailed

description of the solvent - a good description of the interaction is enough• Model: solute (one or more molecules in a cavity) + solvent (mimicked by a

continuous dielectric medium with dielectric constant ε)• Cavity : it should exclude the solvent and contain within its boundaries the largest

possible part of the solute charge distribution• The interaction between solute/solvent is mainly electrostatic (mutual

polarization), formulated mathematically in terms of apparent charges at thesolute/solvent interface (electrostatic interaction solved self-consistently)

11 / 15

The electrostatic problem

We are looking for the solution of a classical electrostatic problem (Poisson), within aQM framework.The charge distribution ρM of the solute, inside the cavity, polarizes the dielectriccontinuum, which in turn polarizes the solute charge distribution (self-consistentprocess). The general Poisson equation:

−∇[ε∇V (r)] = 4πρM (r)

can be simplified to:

−∇2V (r) = 4πρM (r) within C

−ε∇2V (r) = 0 outside C

where C is the portion of space occupied by the cavity. V is the sum of the electrostaticpotential generated by the charge distribution and the reaction potential generated bythe polarization of the dielectric medium.

12 / 15

Solutions of the electrostatic problemSelf-consistent reaction fields methods (SCRF)

• Self-consistent reaction fields (SCRF) methods have different approaches withdifferent definition of the cavity and of the reaction field

• Solutions implemented in Gaussian:

• The Onsager model is the simplest one• The iso density PCM defines the cavity as a surface at constant electronic density• The self-consistent Isodensity polarized continuum model includes the effect of

solvation, accounting for the full coupling between the cavity and the electrondensity

13 / 15

Integral Equation Formalism: IEF PCM (1997)• IEFPCM is the default PCM formulation in Gaussian (from Gaussian G03)• The potentials are redefined in terms of proper Green functions• Only the solute electrostatic potential is required to determine the apparent

surface charge spread on the cavity surface• General approach, but it is not always possible to have analytical Green functions

(anyway they can be built numerically)

14 / 15

Bibliography I

James B. Foresman and Aeleen FrischExploring Chemistry with Electronic Structure Methods - Second EditionGaussian, 1993

James B. Foresman and Aeleen FrischExploring Chemistry with Electronic Structure Methods - Third EditionGaussian, 2015

Tomasi, J., Mennucci, B. and Cammi, R.Quantum Mechanical Continuum Solvation ModelsChemical Reviews, 105(8):2999–3094, 2005

Alecu, I. M. and Zheng, Jingjing and Zhao, Yan and Truhlar, Donald G.Computational Thermochemistry: Scale Factor Databases and Scale Factors forVibrational Frequencies Obtained from Electronic Model ChemistriesJournal of Chemical Theory and Computation, 6(9):287–2887, 2010

15 / 15