1 basic introduction of computational chemistry

7/30/2019 1 Basic Introduction of Computational Chemistry

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Basic Introduction ofComputational Chemistry

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What is Computational Chemistry?

Definition

Some examples

The relation to the real world

Cartoons from "scientist at work; work, scientists, work!"
http://web.archive.org/web/20070125125308/http:/www.nearingzero.net/work/work.htmlhttp://web.archive.org/web/20070125125308/http:/www.nearingzero.net/work/work.html


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Computational Chemistry is

A branch of chemistry

That uses equations encapsulating the behavior ofmatter on an atomistic scale and

Uses computers to solve these equations

To calculate structures and properties

Of molecules, gases, liquids and solidsTo explain or predict chemical phenomena.

See also:

Wikipedia,

http://www.chem.yorku.ca/profs/renef/whatiscc.html [11/10/2010]

http://www.ccl.net/cca/documents/dyoung/topics-orig/compchem.html[11/10/2010]

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http://www.chem.yorku.ca/profs/renef/whatiscc.htmlhttp://www.ccl.net/cca/documents/dyoung/topics-orig/compchem.htmlhttp://www.ccl.net/cca/documents/dyoung/topics-orig/compchem.htmlhttp://www.ccl.net/cca/documents/dyoung/topics-orig/compchem.htmlhttp://www.ccl.net/cca/documents/dyoung/topics-orig/compchem.htmlhttp://www.chem.yorku.ca/profs/renef/whatiscc.html


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Including:

Electron dynamicsTime independent ab initiocalculations

Semi-empiricalcalculations

Classical moleculardynamics

Embedded models

Coarse grained models

Not including:

Quantum chromo-dynamics

Calculations on Jellium

Continuum models

Computational fluid

dynamicsData mining

Rule based derivations

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To explain or predict chemical phenomena:

Phenomenon is any observable occurrenceTherefore computational chemistry has to connect withpractical/experimental chemistry

In many cases fruitful projects live at the interface betweencomputational and experimental chemistry because:

Both domains criticize each other leading to improvedapproaches

Both domains are complementary as results that areinaccessible in the one domain might be easilyaccessible in the other

Agreeing on the problem helps focus the invested effort

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Where do you start?

Selection of energy expressionsHartree-Fock / Density Functional Theory

Moller-Plesset Perturbation TheoryCoupled ClusterQuantum Mechanics / Molecular MechanicsMolecular Mechanics


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You start with

Everything starts with an energy expression

Calculations either minimize to obtain:the ground state

equilibrium geometries

Or differentiate to obtain properties:

Infra-red spectraNMR spectra

Polarizabilities

Or add constraints toOptimize reaction pathways (NEB, string method, ParaReal)

The choice of the energy expression determines theachievable accuracy

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Energy expressions NWChem supports

Effective 1-Electron Models

Hartree-Fock and Density Functional TheoryPlane wave formulation

Local basis set formulation

Correlated Models

Mller-Plesset Perturbation TheoryCoupled Cluster

Combined Quantum Mechanical / MolecularMechanics (QM/MM)

Molecular Mechanics

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Hartree-Fock & Density Functional Theory I

Plane wave & Local basis

The energy expression is derived from a singledeterminant wave function approximation

Replace the exchange with a functional to go from

Hartree-Fock to DFTUse different basis sets for different problems

Plane waves for infinite condensed phase systems

Local basis sets for finite systems


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D=iCiCi

F=h+J(D)+K(D)

+(D)Vxc(D)drFC=C

Hartree-Fock & Density Functional Theory II

Plane wave & Local basis

Minimize energy withrespect to Ci and IIterative processcycling until Self-Consistency

GivesThe total energy E

The molecular orbitals Ci

The orbital energies i

10C

C


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Hartree-Fock & Density Functional Theory III

Local Basis Sets

Largest quantities arethe density, Fock,

overlap, 1-electronmatrices

Memory needed O(N2)Replicated data O(N2)per node

Distributed data O(N2) forwhole calculation

Memory requirements Computational Complexity

Main cost is theevaluation of the 2-

electron integralsTakes O(N2)-O(N4) work

O(N4) for small systems

O(N2) in the large N limit

For large N the linearalgebra becomesdominant at O(N3)

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Hartree-Fock & Density Functional Theory IV

Plane waves

Largest quantities arethe density, Fock,

overlap, 1-electronmatrices

Memory needed O(N2)Replicated data O(N2)per node

Distributed data O(N2) forwhole calculation

Memory requirements ComputationalComplexity

Main cost stems fromthe Fourier transforms

For small systems andlarge processor countsdominated by FFTscosting O(N2*ln(N))

For large systems the

non-local operator andorthogonalization areimportant costing O(N3)

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Mller-Plesset Perturbation Theory I

Assumes that electron correlation effects are small

The Hartree-Fock energy is the 1st order correctedenergy

The 2nd and 3rd order corrected energy can becalculated from the 1st order corrected wavefunction (2N+1 rule)

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Mller-Plesset Perturbation Theory II

The zeroth order energy isthe sum of occupiedorbital energiesThe first order energy is theHartree-Fock energy

The energy correctiongives an estimate of the

interaction of the Hartree-Fock determinant with allsingly and doublesubstituted determinants

The total energy scalescorrectly with system size

It is ill defined if the HOMOand LUMO aredegenerate

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2

2

2

,

,

HF MP

MP

i j occ i j r sr s virt

E E E

ij rsE

http://en.wikipedia.org/wiki/Moller-Plesset_perturbation_theory [11/19/2010]
http://en.wikipedia.org/wiki/Moller-Plesset_perturbation_theoryhttp://en.wikipedia.org/wiki/Moller-Plesset_perturbation_theoryhttp://en.wikipedia.org/wiki/Moller-Plesset_perturbation_theoryhttp://en.wikipedia.org/wiki/Moller-Plesset_perturbation_theory


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Mller-Plesset Perturbation Theory III

The MO basis 2-electron integrals

require the dominantamount of storageThis takes O(N4) storage

Can be reduced toO(N3) by treating theintegral in batches

Memory requirements Computationalcomplexity

Transforming the integralsrequires a summationover all basis functions for

every integralThis takes O(N5) work

If not all transformedintegrals are stored thenthere is an extra cost of

calculating all theintegrals for every batchat O(N4)

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Coupled Cluster I

Sums some corrections to infinite order

Involves singly, doubly, triply-substituted determinantsSimplest form is CCSD

Often necessary to include triply-substituteddeterminants (at least perturbatively), i.e. CCSD(T)

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Coupled Cluster II

The wavefunction is

expressed in anexponential form

The operatorTcontainssingle and doublesubstitution operators

with associatedamplitudes

The vector equation:Solve top two lines for theamplitudes di

r, dijrs

The bottom line gives thetotal energy

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1 2

,

,

00

T HF

r rs

i r i ij r s i j

i occ i j occ

r virt r s virt

rs T T HF

ij

r T T HF

i

HF T T HF

e

T T T

d a a d a a a a

e He

e He

Ee He

http://en.wikipedia.org/wiki/Coupled_cluster[11/19/2010]
http://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_cluster


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Coupled Cluster III

The main objects tostore are thetransformed 2-electron

integrals and theamplitudes

This costs O(N4) storage

Local memory depends

on tile sizes and level oftheoryCCSD O(nt

4)

CCSD(T)O(nt6)

Memory requirements Computational complexity

The main cost are thetensor contractions

For CCSD they can beformulated so that theytake O(N6) work

For CCSD(T) theadditional perturbative

step dominates at O(N7

)

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QM/MM Models I

Describe local chemistry under the influence of an

environmentQuantum region treated with ab-initio method of choice

Surrounded by classical region treated with molecularmechanics

Coupled by electrostatics, constraints, link-atoms, etc.

Crucial part is the coupling of different energyexpressions

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QM/MM Models II

The scheme we are

considering is an hybridscheme (not anadditive scheme likeONIOM)

This scheme is valid for

any situation that theMM and QM methodscan describe, thelimitation lies in theinterface region

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internal external

external

, , ,

, , ,

( ')d '

'

QM MM

QM QM QM

I iIQM

I MM I MM i QMI I i

vdW

E E r R E r R

E E r R E R

Z ZZ rE r

R r R R

E R

http://www.ccl.net/cca/documents/dyoung/topics-orig/qmmm.html [11/20/2010]http://www.salilab.org/~ben/talk.pdf [11/20/2010]http://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.html [11/20/2010]
http://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.htmlhttp://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.htmlhttp://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.htmlhttp://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.htmlhttp://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.htmlhttp://www.ccl.net/cca/documents/dyoung/topics-orig/qmmm.htmlhttp://www.salilab.org/~ben/talk.pdfhttp://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.htmlhttp://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.htmlhttp://www.salilab.org/~ben/talk.pdfhttp://www.ccl.net/cca/documents/dyoung/topics-orig/qmmm.htmlhttp://www.ccl.net/cca/documents/dyoung/topics-orig/qmmm.htmlhttp://www.ccl.net/cca/documents/dyoung/topics-orig/qmmm.html


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QM/MM Models III

Dominated by thememory requirementsof the QM method

See chosen QMmethod, N is now thesize of the QM region


Dominated by thecomplexity of the QMmethod

See chosen QM method,N is now the size of theQM region

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Molecular Mechanics I

Energy of system expressed in terms of relative

positions of atomsThe parameters depend on the atom and theenvironment

A carbon atom is different than a oxygen atom

A carbon atom bound to 3 other atoms is different from one

bound to 4 other atomsA carbon atom bound to hydrogen is different from onebound to fluorine

Etc.

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Molecular mechanics II

Energy terms (parameters)

Bond distances (kAB, r

0

AB)Bond angles (kABC, 0ABC)

Dihedral angles(vABCD;n,

0ABCD)

Van der Waals interactions(AB,AB)

Electrostatic interactions (AB)

The parameters are defined inspecial files

For a molecule the parametersare extracted and stored in the

topology fileThis force field is only valid nearequilibrium geometries

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201

2

,

201

2

, ,

01;2

, , ,

1

2 12 6,

12

,

1 cos

AB AB AB

A B

ABC ABC ABC

A B C

ABCD n ABCD ABCD

A B C D n

AB AB

A B AB AB

AB A B

A B AB

E k r r

k

v n

r r

q q

r

W. Cornell, P. Cieplak, C. Bayly, I. Gould, K. Merz, Jr., D. Ferguson, D. Spellmeyer, T. Fox, J. Caldwell, P.Kollman, J. Am. Chem. Soc., (1996) 118, 2309, http://dx.doi.org/10.1021/ja955032e
http://dx.doi.org/10.1021/ja955032ehttp://dx.doi.org/10.1021/ja955032e


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Molecular Mechanics III

Main data objects are theatomic positions

Storage O(N)


Most terms involve localinteractions between atomsconnected by bonds, thesecost O(N) work

Bond terms

Angle terms

Dihedral angle terms

The remaining two termsinvolve non-local interactions,

cost at worst O(N2

), butimplemented using the particlemesh Ewald summation it costsO(N*log(N))

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Summarizing methods

Method Memory Complexity Strengths

MolecularDynamics

O(N) O(N*ln(N)) Conformational sampling/Freeenergy calculations

Hartree-Fock/DFT O(N2) O(N3) Equilibrium geometries, 1-

electron properties, also excited

states

Mller-Plesset O(N4) O(N5) Medium accuracy correlationenergies, dispersive interactions

Coupled-Cluster O(N4) O(N6)-O(N7) High accuracy correlation

energies, reaction barriers

QM/MM * * Efficient calculations on complex

systems, ground state andexcited state properties

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* Depends on the methods combined in the QM/MM framework.


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What properties might you want to

calculate?

Energies

Equilibrium geometries

Infrared spectra

UV/Vis spectra

NMR chemical shifts

Reaction energies

Thermodynamics

Transition states

Reaction pathways

Polarizabilities


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Energy evaluations

Having chosen an energy expression we cancalculate energies and their differences

Bonding energies

Isomerization energies

Conformational change energies

Identification of the spin state

Electron affinities and ionization potentials

For QM methods the wavefunction and for MMmethods the partial charges are also obtained. Thisallows the calculation of

Molecular potentials (including docking potentials)

Analysis of the charge and/or spin distribution

Natural bond order analysis

Multi-pole moments

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Gradient evaluations

Differentiating the energy with respect to the nuclear

coordinates we get gradients which allows thecalculation ofEquilibrium and transition state geometries

Forces to do dynamics

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Hessian evaluations I

Differentiating the energy twice with respect to the

nuclear coordinates gives the Hessian which allowscalculatingThe molecular vibrational modes and frequencies (if allfrequencies are positive you are at a minimum, if one isnegative you are at a transition state)

Infra-red spectra

Initial search directions for transition state searches

Hessians are implemented for the Hartree-Fock andDensity Functional Theory methods

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Hessian evaluation II

In the effective 1-electron models aperturbed Fock and

density matrix needsto be stored for everyatomic coordinate

The memory requiredis therefore O(N3)


To compute theperturbed densitymatrices a linear system

of dimension O(N2) hasto be solved for everyatomic coordinate

The number ofoperations needed is

O(N5)

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Magnetic properties, e.g. NMR

The chemical shift is calculated as a mixed second

derivative of the energy with respect to the nuclearmagnetic moment and the external magnetic field.

Often the nuclear magnetic moment is treated as aperturbation

Note thatThe paramagnetic and diamagnetic tensors are notrotationally invariant

The total isotropic and an-isotropic shifts are rotationallyinvariant

Requires the solution of the CPHF equations at O(N4)

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http://en.wikipedia.org/wiki/Chemical_shift[11/23/2010]

l i bili i
http://en.wikipedia.org/wiki/Chemical_shifthttp://en.wikipedia.org/wiki/Chemical_shift


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Polarizabilities

Adding an external electric field to the Hamiltonian

and differentiating the energy with respect to thefield strength gives polarizabilityHartree-Fock and DFT

CCSD, CCSDT

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Li I


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Linear response I

Adding a time dependent electric field to the

Hamiltonian, substituting it in the dependentSchrodinger equation, and expanding the time-dependent density in a series an equation for the firstorder correction can be obtained.

This expression is transformed from the time domain

to the frequency domain to obtain an equation forthe excitation energies

Solving this equation for every root of interest has acost of the same order a the corresponding Hartree-Fock or DFT calculation, both in memory

requirements as in the computational complexity.

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http://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdf

Li II
http://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdfhttp://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdfhttp://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdfhttp://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdfhttp://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdfhttp://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdfhttp://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdf


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Linear response II

The equations haveN

occ*N

virtsolutions

Note that the vectorsare normalized butdifferently so than yourusual wavefunction

The orbital energydifference is a mainterm in the excitationenergy

In the case of pureDFT with large

molecules most of theintegrals involving Fxcvanish as this is a localkernel

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http://www.tddft.org/TDDFT2008/lectures/IT2.pdf

* *

,

,

2

1 2

1 2

1 0

0 1

1

,

ia jb ij ab a i H xc

ia jb H xc

xc

A B X X

B A Y Y

X X Y Y

A ia F F jb

B ia F F jb

fF r r

r r

EOM CCSD/CCSD(T)
http://www.tddft.org/TDDFT2008/lectures/IT2.pdfhttp://www.tddft.org/TDDFT2008/lectures/IT2.pdf


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EOM CCSD/CCSD(T)

A Coupled Clustermethod for excitedstates

Depends on the groundstate cluster amplitudes

Memory andcomputationalcomplexity similar tocorresponding CoupledCluster method

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,

,

,

, , ,

T HF

k k

k k

s

k i s i

i s

st

k ij s t i j

i j s t

R e

R r

r a a

r a a a a

T T HF HF

k k ke He R E R


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Which methods do you pick?

Factors in decision making

Available functionality

Accuracy aspects

Th d i i ki f t


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The decision making factors

The method of choice for a particular problem

depends on a number of factors:The availability of the functionalityThe accuracy or appropriateness of the method

The memory requirements and computational cost of themethod

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A il bl f ti lit


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Available functionality

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MM HF/DFT MP2 CC QM/MMEnergy

Gradients

Hessians

Polarizabilities

Excited states

NMR

Gradients and Hessians can always be obtained by

numerical differentiation but this is slow.

http://www.nwchem-sw.org/

A i t f th d
http://www.nwchem-sw.org/http://www.nwchem-sw.org/http://www.nwchem-sw.org/http://www.nwchem-sw.org/


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Appropriateness of method

Too complex a question to answer here in generalFor example consider bond breaking

Molecular Mechanics cannot be used as this is explicitly notpart of the energy expressionHF/DFT can be used but accuracy limited near transition states(unrestricted formulation yields better energies, but often spin-contaminated wavefunctions)Moller-Plesset cannot be used as near degeneracies causesingularities

CCSD or CCSD(T) can be used with good accuracyQM/MM designed for these kinds of calculations of course withthe right choice of QM region

So check your methods before you decide, if necessaryperform some test calculations on a small problem.Often methods that are not a natural fit have been

extended, e.g. dispersion corrections in DFTBottom line: Know your methods!

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W i


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Wrapping up

Further readingAcknowledgements

Questions

Further reading


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Further reading

D. Young, Computational Chemistry: A Practical

Guide for Applying Techniques to Real WorldProblems, Wiley-Interscience, 2001, ISBN:0471333689.

C.J. Cramer, Essentials of Computational Chemistry:Theories and Models, Wiley, 2004, ISBN:0470091827.

R.A.L. Jones Soft Condensed Matter, Oxford

University Press, 2002, ISBN:0198505892.

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Further reading (Books)


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Further reading (Books)

GeneralD. Young, Computational Chemistry: A Practical Guide for Applying Techniques to RealWorld ProblemsC.J. Cramer, Essentials of Computational Chemistry: Theories and Models

F. Jensen, Introduction to Computational Chemistry

Molecular dynamicsFrenkel & Smit, Understanding Molecular Simulation

Allen & Tilldesley, Computer Simulation of Liquids

Leach, MolecularModelling: Principles & Applications

Condensed phaseR.M. Martin, Electronic Structure: basic theory and practical methods

J. Kohanoff, Electronic Structure Calculations for Solids and Molecules

D. Marx, J. Hutter, Ab Initio Molecular Dynamics

Quantum chemistryOstlund & Szabo, Modern Quantum Chemistry

Helgaker, Jorgensen, Olsen, Molecular Electronic Structure Theory

McWeeny, Methods of Molecular Quantum Chemistry

Parr & Yang, Density Functional Theory of Atoms & MoleculesMarques et al, Time-Dependent Density Functional Theory

OtherJanssen & Nielsen, Parallel Computing in Quantum Chemistry

Shavitt & Bartlett, Many-Body Methods in Chemistry and Physics

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Acknowledgement


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Acknowledgement

This work was produced using EMSL, a national

scientific user facility sponsored by the Departmentof Energy's Office of Biological and EnvironmentalResearch and located at Pacific Northwest NationalLaboratory.

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Questions?


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Questions?

1 basic introduction of computational chemistry

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