Molecule properties from QM modeling

Shyue Ping Ong

Our journey so far

Schrodinger Equation

Variational Approaches

Hartree Fock

Including Correlation with

Hartree Fock

Density Functional

Theory

Local density approximation

Generalized gradient

approximation

Hybrids

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It’s time to see what we can do with these

The Materials World

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Molecules

Isolated gas phase

Typically use localized basis functions, e.g.,

Gaussians

Everything else (liquids,

amorphous solids, etc.)

Too complex for direct QM!

(at the moment)

But can work reasonable

models sometimes

Crystalline solids

Periodic infinite solid

Plane-wave approaches

Overview

In this lecture, we will •  Survey the study of properties of isolated

molecules using quantum mechanical approaches.

•  Connect calculations with real world properties •  Discuss performance and accuracy

Lab 1: Study of ammonia formation using QM

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What do you get from QM?

Energies Geometries Charge densities and spectroscopic properties

And their derivatives…

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Energies and eigenvalues

Most direct output from QM calculations Accuracy have been discussed in previous lectures

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Vibrational frequencies and energies

Harmonic oscillator assumption To obtain the force constants, one simply needs to calculate the 2nd derivative of the energy with respect to bond stretching at equilibrium bond geometry Can be done analytically for HF, MP2, DFT, CISD, CCSD

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E = n+ 12

!

"#

$

%&hω where ω =

12π

kµ

where k is the force constants

Scaling factors for vibrational frequencies

To account for systematic errors in predicted vibrational frequencies

E.g., HF overemphasizes bonding and all force constants (and frequencies) are too large

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Ensemble thermodynamic ensembles

QM gives the single molecule energies Question: How do we get ensemble thermodynamic variables from single-molecule calculations?

Answer: Statistical mechanics

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A brief recap of statistical mechanics

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Z(N,V,T ) = e−Ei (N ,V )kBT

i∑

U = kBT2 ∂ lnZ

∂T$

%&

'

()N ,V

H =U +PV

s = kB lnZ + kB∂ lnZ∂T

$

%&

'

()N ,V

G = H −TS

Assumption: Ideal gas molecules

Since ideal gas molecules do not interact,

The molecular partition function can be further broken down into separable components

Combining the results, we have

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Z(N,V,T ) = z(V,T )N

N! where z(V,T ) is the molecular partition function.

z(V,T ) = zelec (T )ztrans (V,T )zrot (T )zvib(T )

ln Z(N,V,T )( ) = N zelec (T )+ ztrans (V,T )+ zrot (T )+ zvib(T )[ ]− N lnN + N

Components of the partition function

Electronic •  Typically, excited states are much higher in energy and make no significant

contribution to partition function below a few 1000K. => Just the electronic energy from QM.

•  If there is a non-singlet ground state, there are contributions to the electronic entropy.

Translation (Particle in box)

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ztrans (V,T ) =2πMk BT

h2!

"#

$

%&

32V

Utrans =32RT

Strans0 = R ln 2πMk BT

h2!

"#

$

%&

32 VNA

'

(

))

*

+

,,+52

-

./

0/

1

2/

3/

Components of the partition function

Vibrational •  Based on quantum mechanical harmonic oscillator assumption

(3N – 6 degrees of freedom)

Rotational

•  Linear and non-linear molecules to be treated separately •  Refer to statistical mechanics textbook

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zvib(T ) =1

1− e−hω /kBT

Uvib =1

1− e−hωi /kBTi=1

3N−6

∏

Svib0 = R hωi

kBT (ehωi /kBT −1)

− ln(1− e−hωi /kBT )#

$%

&

'(

i=1

3N−6

∑

Typical calculation procedure for enthalpies

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Geometry optimization

(GO)

•  Typically at a lower level of theory and smaller basis set

Frequency calculation

•  Same level of theory as GO

•  Obtain vibrational and other contributions to free energy

SCF energy calculation

•  Higher level of theory and basis set

Selecting model chemistries

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Foresman, J. B.; Frisch, Ae. Exploring Chemistry With Electronic Structure Methods: A Guide to Using Gaussian; Gaussian, 1996.

Practical reaction calculations

Let’s say we are interested in calculating the following reaction energies from QM

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Reaction 1N2 (g)+3H2 (g)→ 2NH3(g)

Reaction 2C(s)+O2 (g)→ 2CO2 (g)

This one is easy. I just calculate the energies in the gas state for each of the molecules with the statistical corrections! -> Subject of Lab 1

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ΔH 0f ,298(M ) = E(M )+ ZPE(M )+[H298(M )−H0 (M )]

− E(Xz )+[H298(Xz )−H0 (Xz )]{ }z

atoms

∑ + ΔH 0f ,298

z

atoms

∑ (Xz )

Ionization energies and electron affinities

Koopman’s Theorem •  HOMO energy as estimate of vertical IE fairly reasonable due to

canceling of basis set incompleteness and correlation errors in Hartree-Fock

•  Though a corollary of Koopman exists for DFT for the exact xc functional, in practice eigenvalues from inexact DFT are poor estimates.

ΔSCF

•  Calculate energy of molecule in neutral and positively / negatively charged

•  Generally works well if diffuse functions are used to model ions with diffuse electron clouds.

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Charge distribution properties

Multipole moments Partial atomic charges

•  Class II charges – determined by partitioning of wave functions (a somewhat arbitrary process)

•  Mulliken approach – partition according to degree atomic orbitals contribute to wave function

•  Lowdin – Transform AO basis functions to orthonormal set •  Natural population analysis (NPA) – Orthogonalization in four-

step process to render electron density as compact as possible before Mulliken analysis

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xkylzm = Zixik yi

lzim

i

atoms

∑ − ψ(r) xik yi

lzim

i

electrons

∑∫ ψ(r)dr

NMR spectral properties

General recommendation is very large basis sets (at least triple-ζ) and lots of diffuse and polarization functions Not possible to predict chemical shift for nuclei of heavy atoms with effective core potentials For molecules comprising first row atoms, heavy-atom chemical shifts can be obtained with a fair degree of accuracy, even with HF (though DFT and MP2 fares much better).

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