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Introduction to Computational Chemistry
Alexander B. Pacheco
User Services ConsultantLSU HPC & LONIsys-help@loni.org
LONI Worshop SeriesTulane University, New Orleans
March 25, 2011
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Outline
1 Introduction
2 Ab Initio Methods
3 Density Functional Theory
4 Semi-empirical Methods
5 Basis Sets
6 Molecular Mechanics
7 Quantum Mechanics/Molecular Mechanics (QM/MM)
8 Molecular Dynamics
9 Computational Chemistry Programs
10 Example Jobs
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What is Computational Chemistry
Computational Chemistry is a branch of chemistry thatuses computer science to assist in solving chemicalproblems.Incorporates the results of theoretical chemistry intoefficient computer programs.Application to single molecule, groups of molecules, liquidsor solids.Calculates the structure and properties of interest.Computational Chemistry Methods range from
1 Highly accurate (Ab-initio,DFT) feasible for small systems2 Less accurate (semi-empirical)3 Very Approximate (Molecular Mechanics) large systems
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What is Computational Chemistry
Theoretical Chemistry: broadly can be divided into twomain categories
1 Static Methods⇒ Time-Independent Schrödinger Equation
♦ Quantum Chemical/Ab Initio /Electronic Structure Methods♦ Molecular Mechanics
2 Dynamical Methods⇒ Time-Dependent SchrödingerEquation♦ Classical Molecular Dynamics♦ Semi-classical and Ab-Initio Molecular Dynamics
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Ab Initio Methods
Ab Initio meaning "from first principles" methods solve the Schrödingerequation and does not rely on empirical or experimental data.
Begining with fundamental and physical properties, calculate howelectrons and nuclei interact.
The Schrödinger equation can be solved exactly only for a few systems
♦ Particle in a Box♦ Rigid Rotor♦ Harmonic Oscillator♦ Hydrogen Atom
For complex systems, Ab Initio methods make assumptions to obtainapproximate solutions to the Schrödinger equations and solve itnumerically.
"Computational Cost" of calculations increases with the accuracy of thecalculation and size of the system.
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Ab Initio Methods
What can we predict with Ab Initio methods?Molecular Geometry: Equilibrium and Transition State
Dipole and Quadrupole Moments and polarizabilities
Thermochemical data like Free Energy, Energy of reaction.
Potential Energy surfaces, Barrier heights
Reaction Rates and cross sections
Ionization potentials (photoelectron and X-ray spectra) and Electronaffinities
Frank-Condon factors (transition probabilities, vibronic intensities)
Vibrational Frequencies, IR and Raman Spectra and Intensities
Rotational spectra
NMR Spectra
Electronic excitations and UV-VIS spectra
Electron density maps and population analyses
Thermodynamic quantities like partition function
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Ab Initio Methods
Ab Initio TheoryBorn-Oppenheimer Approximation: Nuclei are heavier than electronsand can be considered stationary with respect to electrons. Also knowas "clamped nuclei" approximations and leads to idea of potentialsurface
Slater Determinants: Expand the many electron wave function in termsof Slater determinants.
Basis Sets: Represent Slater determinants by molecular orbitals, whichare linear combination of atomic-like-orbital functions i.e. basis sets
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Born-Oppenheimer Approximation I
Solve time-independent Schrödinger equation
HΨ = EΨ
For many electron system:
H = −~2
2
∑α
∇2α
Mα︸ ︷︷ ︸Tn
− ~2
2me
∑i
∇2i︸ ︷︷ ︸
Te
+∑α>β
e2ZαZβ4πε0Rαβ︸ ︷︷ ︸
Vnn
−∑α,i
e2Zα4πε0Rαi︸ ︷︷ ︸
Ven
+∑i>j
e2
4πε0rij︸ ︷︷ ︸Vee︸ ︷︷ ︸
V
The wave function Ψ(R, r) of the many electron molecule is a function ofnuclear (R) and electronic (r) coordinates.
Motion of nuclei and electrons are coupled.
However, since nuclei are much heavier than electrons, the nucleiappear fixed or stationary.
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Born-Oppenheimer Approximation II
Born-Oppenheimer Approximation: Separate electronic and nuclearmotion:
Ψ(R, r) = ψe(r; R)ψn(R)
Solve electronic part of Schrödinger equation
Heψe(r; R) = Eeψe(r; R)
BO approximation leads tothe concept of potentialenergy surface
V(R) = Ee + Vnn
−200
−100
0
100
200
300
1 2 3 4 5V
(kca
l/m
ol)
RHO−H(a0)
De
Re
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Potential Energy Surfaces
The potential energy surface (PES) is multi-dimensional (3N − 6 fornon-linear molecule and 3N − 5 for linear molecule)The PES contains multiple minima and maxima.Geometry optimization search aims to find the global minimum of thepotential surface.Transition state or saddle point search aims to find the maximum of thispotential surface, usually along the reaction coordinate of interest.
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Wavefunction Methods I
The electronic Hamiltonian (in atomic units, ~,me, 4πε0, e = 1) to besolved is
He = −12
∑i
∇2i −
∑α,i
ZαRiα
+∑i>j
1rij
+∑α>β
ZαZβRαβ
Calculate electronic wave function and energy
Ee =〈ψe | He | ψe〉〈ψe | ψe〉
The total electronic wave function is written as a Slater Determinant ofthe one electron functions, i.e. molecular orbitals, MO’s
ψe =1√N!
∣∣∣∣∣∣∣∣φ1(1) φ2(1) · · · φN(1)φ1(2) φ2(2) · · · φN(2)· · · · · · · · · · · ·φ1(N) φ2(N) · · · φN(N)
∣∣∣∣∣∣∣∣
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Wavefunction Methods II
MO’s are written as a linear combination of one electron atomicfunctions or atomic orbitals (AO’s)
φi =N∑µ=1
cµiχµ
cµi ⇒ MO coefficientsχµ ⇒ atomic basis functions.
Obtain coefficients by minimizing the energy via Variational Theorem.
Variational Theorem: Expectation value of the energy of a trialwavefunction is always greater than or equal to the true energy
Ee = 〈ψe | He | ψe〉 ≥ ε0
Increasing N ⇒ Higher quality of wavefunction⇒ Higher computationalcost
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Ab Initio Methods
The most popular classes of ab initio electronic structuremethods:
Hartree-Fock methods
♦ Hartree-Fock (HF)� Restricted Hartree-Fock (RHF): singlets� Unrestricted Hartree-Fock (UHF): higher multiplicities� Restricted open-shell Hartree-Fock (ROHF)
Post Hartree-Fock methods
♦ Møller-Plesset perturbation theory (MPn)♦ Configuration interaction (CI)♦ Coupled cluster (CC)
Multi-reference methods
♦ Multi-configurational self-consistent field (MCSCF)♦ Multi-reference configuration interaction (MRCI)♦ n-electron valence state perturbation theory (NEVPT)♦ Complete active space perturbation theory (CASPTn)
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Hartree-Fock
1 Wavefunction is written as a single determinant
Ψ = det(φ1, φ2, · · ·φN)
2 The electronic Hamiltonian can be written as
H =∑
i
h(i) +∑i>j
v(i, j)
where h(i) = −12∇2
i −∑i,α
Zαriα
and v(i, j) =1rij
3 The electronic energy of the system is given by:
E = 〈Ψ|H|Ψ〉
4 The resulting HF equations from minimization of energy by applying ofvariational theorem:
f (x1)χi(x1) = εiχi(x1)
where εi is the energy of orbital χi and the Fock operator f , is defined as
f (x1) = h(x1) +∑
j
[Jj(x1)− Kj(x1)
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Hartree-Fock
1 Jj ⇒ Coulomb operator⇒ average potential at x due to chargedistribution from electron in orbital χi defined as
Jj(x1)χi(x1) =
[∫χ∗j (x2)χj(x2)
r12dx2
]χi(x1)
2 Kj ⇒ Exchange operator⇒ Energy associated with exchange ofelectrons⇒ No classical interpretation for this term.
Kj(x1)χi(x1) =
[∫χ∗j (x2)χi(x2)
r12dx2
]χj(x1)
3 The Hartree-Fock equation are solved numerically or in a spacespanned by a set of basis functions (Hartree-Fock-Roothan equations)
χi =K∑µ=1
Cµiχµ∑ν
FµνCνi = εi
∑ν
SµνCνi
FC = SCε
Sµν =
∫dx1χ
∗µ(x1)χν(x1)
Fµν =
∫dx1χ
∗µ(x1)f (x1)χν(x1)
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Hartree-Fock
1 The Hartree-Fock-Roothan equation is a pseudo-eigenvalue equation2 C’s are the expansion coefficients for each orbital expressed as a linear
combination of the basis function.3 Note: C depends on F which depends on C⇒ need to solve self-consistently.4 Starting with an initial guess orbitals, the HF equations are solved iteratively or
self consistently (Hence HF procedure is also known as self-consistent field orSCF approach) obtaining the best possible orbitals that minimize the energy.
SCF procedure
1 Specify molecule, basis functions and electronic state of interest2 Form overlap matrix S3 Guess initial MO coefficients C4 Form Fock Matrix F5 Solve FC = SCε
6 Use new MO coefficients C to build new Fock Matrix F7 Repeat steps 5 and 6 until C no longer changes from one iteration to the next.
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What are Post Hartree-Fock Methods
1 In Hartree-Fock theory, electron motions of independent of each other i.e.uncorrelated.
2 However, this is not true. For two electrons with same spin| Ψ1(r1)α(ω1)Ψ2(r2)α(ω2)〉, the probability of finding electron 1 at r1 and electron2 at r2
P(r1, r2)dr1dr2 =12
(|Ψ1(r1)|2|Ψ2(r2)|2 + |Ψ1(r2)|2|Ψ2(r1)|2
− [Ψ∗1 (r1)Ψ2(r1)Ψ∗2 (r2)Ψ1(r2)
+ Ψ∗2 (r1)Ψ1(r1)Ψ∗1 (r2)Ψ2(r2)]) dr1dr2
Now P(r1, r1) = 0⇒ No two electrons with same spins can be at the same place⇒ "Fermi hole"
3 Same-spin electrons are correlated while different spin electrons are not.4 Energy difference between HF energy and the true energy is the correlation
energy
Ecorr = E0 − EHF
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♦ Methods that improve the Hartree-Fock results by accounting for the correlation energyare known as Post Hartree-Fock methods
♦ The starting point for most Post HF methods is the Slater Determinant obtain fromHartree-Fock Methods.
♦ Configuration Interaction (CI) methods: Express the wavefunction as a linearcombination of Slater Determinants with the coeffcients obtained variationally
|Ψ〉 =∑
I
cI |ΨI〉
♦ Many Body Perturbation Theory: Treat the HF determinant as the zeroth order solutionwith the correlation energy as a pertubation to the HF equation.
H = H0 + λH′; εi = E(0)i + λE(1)
i + λ2E(2)i + · · ·
|Ψi〉 = |Ψ(0)i 〉+ λ|Ψ(1)
i 〉+ λ2|Ψ(2)i 〉 · · ·
♦ Coupled Cluster Theory:The wavefunction is written as an exponential ansatz
|Ψ〉 = eT |Ψ0〉
where |Ψ0〉 is a Slater determinant obtained from HF calculations and T is an excitationoperator which when acting on |Ψ0〉 produces a linear combination of excited Slaterdeterminants.
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Scaling
Scaling Behavior Method(s)
N4 HF
N5 MP2
N6 MP3, CISD, CCSD, QCISD
N7 MP4, CCSD(T), QCISD(T)
N8 MP5, CISDT, CCSDT
N9 MP6
N10 MP7, CISDTQ, CCSDTQ
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Density Functional Theory I
Density Functional Theory (DFT) is an alternative to wavefunction basedelectronic structure methods of many-body systems such as Hartree-Fock andPost Hartree-Fock.
In DFT, the ground state energy is expressed in terms of the total electrondensity.
ρ0(r) = 〈Ψ0|ρ|Ψ0〉
We again start with Born-Oppenheimer approximation and write the electronicHamiltonian as
H = F + Vext
where F is the sum of the kinetic energy of electrons and the electron-electroninteraction and Vext is some external potential.
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Density Functional Theory II
Modern DFT methods result from the Hohenberg-Kohn theorem1 The external potential Vext, and hence total energy is a unique functional of
the electron density ρ(r)
Energy =〈Ψ | H | Ψ〉〈Ψ | Ψ〉 ≡ E[ρ]
2 The ground state energy can be obtained variationally, the density thatminimizes the total energy is the exact ground state density
E[ρ] > E[ρ0], ifρ 6= ρ0
If density is known, then the total energy is:
E[ρ] = T[ρ] + Vne[ρ] + J[ρ] + Enn + Exc[ρ]
where
Enn[ρ] =∑A>B
ZAZB
RABVne[ρ] =
∫ρ(r)Vext(r)dr
J[ρ] =12
∫ρ(r1)ρ(r2)
r12dr1dr2
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Density Functional Theory III
If the density is known, the two unknowns in the energy expression are thekinetic energy functional T[ρ] and the exchange-correlation functional Exc[ρ]
To calculate T[ρ], Kohn and Sham introduced the concept of Kohn-Sham orbitalswhich are eigenvectors of the Kohn-Sham equation(
−12∇2 + veff(r)
)φi(r) = εiφi(r)
Here, εi is the orbital energy of the corresponding Kohn-Sham orbital, φi, and thedensity for an ”N”-particle system is
ρ(r) =
N∑i
|φi(r)|2
The total energy of a system is
E[ρ] = Ts[ρ] +
∫dr vext(r)ρ(r) + VH[ρ] + Exc[ρ]
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Density Functional Theory IV
Ts is the Kohn-Sham kinetic energy which is expressed in terms of the Kohn-Sham orbitalsas
Ts[ρ] =N∑
i=1
∫dr φ∗i (r)
(−
12∇2)φi(r)
vext is the external potential acting on the interacting system (at minimum, for a molecularsystem, the electron-nuclei interaction), VH is the Hartree (or Coulomb) energy,
VH =12
∫drdr′
ρ(r)ρ(r′)|r − r′|
and Exc is the exchange-correlation energy.
The Kohn-Sham equations are found by varying the total energy expression with respectto a set of orbitals to yield the Kohn-Sham potential as
veff(r) = vext(r) +
∫ρ(r′)|r − r′|
dr′ +δExc[ρ]
δρ(r)
where the last term vxc(r) ≡δExc[ρ]
δρ(r)is the exchange-correlation potential.
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Density Functional Theory V
The exchange-correlation potential, and the corresponding energy expression,are the only unknowns in the Kohn-Sham approach to density functional theory.
There are many ways to approximate this functional Exc, generally divided intotwo separate terms
Exc[ρ] = Ex[ρ] + Ec[ρ]
where the first term is the exchange functional while the second term is thecorrelation functional.
Quite a few research groups have developed the exchange and correlationfunctionals which are fit to empirical data or data from explicity correlatedmethods.
Popular DFT functionals (according to a recent poll)
♦ PBE0 (PBEPBE), B3LYP, PBE, BP86, M06-2X, B2PLYP, B3PW91, B97-D,M06-L, CAM-B3LYP
� http://www.marcelswart.eu/dft-poll/index.html
� http://www.ccl.net/cgi-bin/ccl/message-new?2011+02+16+009
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Semi-empirical Methods
Semi-empirical quantum methods:♦ Represents a middle road between the mostly qualitative results
from molecular mechanics and the highly computationallydemanding quantitative results from ab initio methods.
♦ Address limitations of the Hartree-Fock claculations, such asspeed and low accuracy, by omitting or parametrizing certainintegrals
integrals are either determined directly from experimental data orcalculated from analytical formula with ab initio methods or fromsuitable parametric expressions.
Integral approximations:♦ Complete Neglect of Differential Overlap (CNDO)♦ Intermediate Neglect of Differential Overlap (INDO)♦ Neglect of Diatomic Differential Overlap (NDDO) ( Used by PM3,
AM1, ...)
Semi-empirical methods are fast, very accurate when applied to moleculesthat are similar to those used for parametrization and are applicable to verylarge molecular systems.
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Heirarchy of Methods
Hartree-‐Fock HF-‐SCF
Excita1on Hierarchy CIS,CISD,CISDT CCS,CCSD,CCSDT
Mul1configura1onal HF MCSCF,CASSCF
Perturba1on Hierarchy MP2,MP3,MP4
Mul1reference Perturba1on CASPT2, CASPT3
Full CI
Semi-‐empirical methods CNDO,INDO,AM1,PM3
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Basis Sets I
Slater type orbital (STO) or Gaussian type orbital (GTO) to describe theAO’s
χSTO(r) = xlymzne−ζr
χGTO(r) = xlymzne−ξr2
where L = l + m + n is the total angular momentun and ζ, ξ are orbitalexponents.
0
0.5
1
0 1 2 3 4 5
Am
pli
tud
e
r (a0)
STOGTO
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Basis Sets II
Why STOCorrect cups at r → 0
Desired decay at r →∞Correctly mimics H orbitals
Natural Choice for orbitals
Computationally expensive tocompute integrals andderivatives.
Why GTOWrong behavior at r → 0 andr →∞Gaussian × Gaussian =Gaussian
Analytical solutions for mostintegrals and derivatives.
Computationally less expensivethan STO’s
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Basis Sets III
Pople family basis set1 Minimal Basis: STO-nG
♦ Each atom optimized STO is fit with n GTO’s♦ Minimum number of AO’s needed
2 Split Valence Basis: 3-21G,4-31G, 6-31G
♦ Contracted GTO’s optimized per atom.♦ Valence AO’s represented by 2 contracted GTO’s
3 Polarization: Add AO’s with higher angular momentum (L)
♦ 3-21G* or 3-21G(d),6-31G* or 6-31G(d),6-31G** or6-31G(d,p)
4 Diffuse function: Add AO with very small exponents for systems withdiffuse electron densities
♦ 6-31+G*, 6-311++G(d,p)
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Basis Sets IV
Correlation consistent basis set♦ Family of basis sets of increasing sizes.
♦ Can be used to extrapolate basis set limit.
♦ cc-pVDZ: Double Zeta(DZ) with d’s on heavy atoms, p’s on H
♦ cc-pVTZ: triple split valence with 2 sets of d’s and 1 set of f’s on heavyatom, 2 sets of p’s and 1 set of d’s on H
♦ cc-pVQZ, cc-pV5Z, cc-pV6Z
♦ can be augmented with diffuse functions: aug-cc-pVXZ (X=D,T,Q,5,6)
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Basis Sets V
Pseudopotentials or Effective Core Potentials♦ All Electron calculations are prohibitively expensive.
♦ Only valence electrons take part in bonding interaction leaving coreelectrons unaffected.
♦ Effective Core Potentials (ECP) a.k.a Pseudopotentials describeinteractions between the core and valence electrons.
♦ Only valence electrons explicitly described using basis sets.
♦ Pseudopotentials commonly used
� Los Alamos National Laboratory: LanL1MB and LanL2DZ� Stuttgard Dresden Pseudopotentials: SDDAll can be used.� Stevens/Basch/Krauss ECP’s: CEP-4G,CEP-31G,CEP-121G
♦ Pseudopotential basis are "ALWAYS" read in pairs
� Basis set for valence electrons� Parameters for core electrons
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Molecular Mechanics
The potential energy of all systems in molecular mechanics iscalculated using force fields.
Molecular mechanics can be used to study small molecules as well aslarge biological systems or material assemblies with many thousands tomillions of atoms.
All-atomistic molecular mechanics methods have the followingproperties:
♦ Each atom is simulated as a single particle♦ Each particle is assigned a radius (typically the van der Waals
radius), polarizability, and a constant net charge (generally derivedfrom quantum calculations and/or experiment)
♦ Bonded interactions are treated as "springs" with an equilibriumdistance equal to the experimental or calculated bond length
The exact functional form of the potential function, or force field,depends on the particular simulation program being used.
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Molecular Mechanics
General form of MolecularMechanics equations
E = Ebond + Eangle + Edihedral + EvdW + Eelec
=12
∑bonds
Kb(b− b0)2
+12
∑angles
Kθ(θ − θ0)2
+12
∑dihedrals
Kφ [1 + cos(nφ)]2
+∑
non−bonds
[(σ
r
)12−(σ
r
)6]
+q1q2
Dr
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QM/MM
Quantum Mechanics(QM): Accurate, expensive (O(N4)), suitable forsmall systems.
Molecular Mechanics(MM): Approximate, does not treat electronsexplicitly, suitable for large systems such as enzymes and proteins,cannot simulate bon breaking/forming
What do we do if we want simulate chemical reaction in large systems?
Methods that combine QM and MM are the solution.
Such methods are called Hybrid QM/MM methods.
The basic idea is to partition the system into two (or more) parts1 The region of chemical interest is treated using accurate QM
methods eg. active site of an enzyme.2 The rest of the system is treated using MM or less accurate QM
methods such as semi-empirical methods or a combination of thetwo.
HTotal = HQM + HMM + HintQM−MM
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QM/MM
ONIOM: divide the system into a real (full) system and the modelsystem. Treat the model system at high and low level. The total energyof the system is given by
E = E(low, real) + E(high,model)− E(low,model)
Empirical Valence Bond: Treat any point on a reaction surface as acombination of two or more valence bond structures
Effective Fragment Potential: add fragments to a standard QMtreatment, which are fully polarizable and are parameterized fromseparate ab initio calculations.
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Molecular Dynamics
Why Molecular Dynamics?
Electronic Structure Methods are applicable to systems in gas phase under lowpressure (vaccum).
Majority of chemical reactions take place in solution at some temperature withbiological reactions usually at specific pH’s.
Calculating molecular properties taking into account such environmental effectswhich can be dynamical in nature are not adequately described by electronicstructure methods.
Molecular Dynamics
Generate a series of time-correlated points in phase-space (a trajectory).
Propagate the initial conditions, position and velocities in accordance withNewtonian Mechanics. F = ma = −∇V
Fundamental Basis is the Ergodic Hypothesis: the average obtained byfollowing a small number of particles over a long time is equivalent to averagingover a large number of particles for a short time.
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Molecular Dynamics Theory
Solve the time-dependent Schrödinger equation
ı~ ∂∂t
Ψ(R, r, t) = HΨ(R, r, t)
with
Ψ(R, r, t) = χ(R, t)Φ(r, t)
and
H = −∑
I
~2
2MI∇2
I +−~2
2me∇2
i + Vn−e(r,R)︸ ︷︷ ︸He(r,R)
Obtain coupled equations of motion for electrons and nuclei:Time-Dependent Self-Consistent Field (TD-SCF) approach.
ı~∂Φ
∂t=
[−∑
i
~2
2me∇2
i + 〈χ|Vn−e|χ〉
]Φ
ı~∂χ∂t
=
[−∑
I
~2
2MI∇2
I + 〈Φ|He|Φ〉
]χ
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Define nuclear wavefunction as
χ(R, t) = A(R, t) exp [iS(R, t)/~]
where A and S are real.Solve the time-dependent equation for nuclear wavefunction and takeclassical limit (~→ 0) to obtain
∂S∂t
+∑
I
~2
2MI(∇IS)2 + 〈Φ|He|Φ〉 = 0
an equation that is isomorphic with the Hamilton-Jacobi equation withthe classical Hamilton function given by
H({RI}, {PI}) =∑
I
~2
2MIP2
I + V({RI})
where
PI ≡ ∇IS and V({RI}) = 〈Φ|He|Φ〉
Obtain equations of nuclear motion from Hamilton’s equationdPI
dt= − dH
dRI⇒ MRI = −∇IV
dRI
dt=
dHdPI
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Replace nuclear wavefunction by delta functions centered on nuclearposition to obtain
i~∂Φ
∂t= He(r, {RI})Φ(r; {RI}, t)
This approach of simultaneously solving the electronic and nucleardegrees of freedom by incorporating feedback in both directions isknown as Ehrenfest Molecular Dynamics.
Expand Φ in terms of many electron wavefunctions or determinants
Φ(r; {RI}, t) =∑
i
ci(t)Φi(r; {RI})
with matrix elements
Hij = 〈Φi|He|Φj〉
Inserting Φ in the TDSE above, we get
ı~ci(t) = ci(t)Hii − ı~∑
I,i
RIdijI
with non-adiabtic coupling elements given by
dijI (RI) = 〈Φi|∇I |Φj〉
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Upto this point, no restriction on the nature of Φi i.e. adiabatic ordiabatic basis has been made.
Ehrenfest method rigorously includes non-adiabtic transitions betweenelectronic states within the framework of classical nuclear motion andmean field (TD-SCF) approximation to the electronic structure.
Now suppose, we define {Φi} to be the adiabatic basis obtained fromsolving the time-independent Schrödinger equation,
He(r, {RI})Φi(r; {RI}) = Ei({RI})Φi(r; {RI})
The classical nuclei now move along the adiabatic orBorn-Oppenheimer potential surface. Such dynamics are commonlyknown as Born-Oppenheimer Molecular Dynamics or BOMD.
If we restrict the dynamics to only the ground electronic state, then weobtain ground state BOMD.
If the Ehrenfest potential V({RI}) is approximated to a global potentialsurface in terms of many-body contributions {vn}.
V({RI}) ≈ Vapproxe (R) =
N∑I=1
v1(RI) +
N∑I>J
v2(RI ,RJ) +
N∑I>J>K
v3(RI ,RJ ,RK) + · · ·
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Thus the problem is reduced to purely classical mechanics once the{vn} are determined usually Molecular Mechanics Force Fields. Thisclass of dynamics is most commonly known as Classical MolecularDynamics.If we follow Lagrangian mechanics and write down the Lagrangian forthe system of interest.
L = T − V
Corresponding Newton’s equation of motion are then obtained from theassociated Euler-Lagrange equations,
ddt∂L∂RI
=∂L∂RI
The Lagrangian for ground state BOMD is
LBOMD =∑
I
12
MIR2I −min
Φ0〈Φ0|He|Φ0〉
and equations of motions
MIRI =ddt∂LBOMD
∂RI=∂LBOMD
∂RI= −∇I min
Φ0〈Φ0|He|Φ0〉
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Extended Lagrangian Molecular Dynamics (ELMD): Extend theLagrangian by adding kinetic energy of fictitious particles, i.e. treatquantum degrees of freedom as classical particles and obtain theirequation of motions from Euler-Lagrange equations.Car-Parrinello Molecular Dynamics (CPMD):
LCPMD =∑
I
12
MIR2I +
∑i
12µi〈φi|φi〉 − 〈Φ0|He|Φ0〉+ constraints
Atom centered Density Matrix Propagation (ADMP):
LADMP =12
Tr(VT MV) +12µTr(WW)− E(R,P)− Tr[Λ(PP− P)]
whereddt
P = W
curvy-steps ELMD (csELMD):
LcsELMD =∑
I
12
MIR2I +
12µ∑i<j
∆ij − Tr[(h +12
H)P]− Vnuc
where
P(λ) = eλ∆P(0)e−λ∆
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Molecular Dynamics: Methods and Programs
Electronic energy obtained fromMolecular Mechanics⇒ Classical Molecular Dynamics
1 LAMMPS2 NAMD3 Amber4 Gromacs
Ab-Initio Methods⇒ Quantum or Ab-Initio MolecularDynamics
1 Born-Oppenheimer Molecular Dynamics: Gaussian,GAMESS
2 Extended Lagrangian Molecular Dynamics: VASP, CPMD,Gaussian (ADMP), NWCHEM(CPMD), QChem (curvy-stepsELMD)
3 Time Dependent Hartree-Fock and Time Dependent DensityFunctional Theory: Gaussian, GAMESS, NWCHEM, QChem
4 Multiconfiguration Time Dependent Hartree(-Fock),MCTDH(F)
5 Non-Adiabatic and Ehrenfest Molecular Dynamics, MultipleSpawning, Trajectory Surface Hopping
6 Wavepacket Methods: Gaussian (QWAIMD)
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Classical Molecular DynamicsAdvantages
1 Large Biological Systems2 Long time dynamics
Disadvantages1 Cannot describe Quantum Nuclear Effects
Ab Initio and Quantum DynamicsAdvantages
1 Quantum Nuclear EffectsDisadvantages
1 ∼ 100 atoms2 Full Quantum Dynamics ie treating nuclei quantum
mechanically: less than 10 atoms3 Picosecond dynamics at best
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Software QB Eric Louie Oliver Painter Poseidon Philip Tezpur
Amber X X X X X X X XDesmond XDL_Poly X X X X X X X XGromacs X X X X X X X XLAMMPS X X X X X X X X
NAMD X X X X X X XOpenEye X X X X X X X X
CPMD X X X X X X XGAMESS X X X X X X X XGaussian X X X X X XNWCHEM X X X X X X XPiny_MD X X X X X X X X
Software Bluedawg Ducky Lacumba Neptune Zeke Pelican
Amber X X XGromacs X X X X X XLAMMPS X X X X X X
NAMD X X X X XCPMD X X X X X
Gaussian X X X X XNWCHEM X X X X X XPiny_MD X X X X X X
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Computational Chemistry Programs
Commercial Software: Q-Chem, Jaguar,CHARMMGPL/Free Software: ACES, ABINIT, Octopushttp://en.wikipedia.org/wiki/Quantum_chemistry_computer_programs
http://www.ccl.net/chemistry/links/software/index.shtml
http://www.redbrick.dcu.ie/~noel/linux4chemistry/
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Job Types and Keywords I
Job Type Gaussian GAMESS NWCHEM# keyword runtyp= task
Energy sp energy energyForce force gradient gradient
Geometry optimization opt optimize optimizeTransition State opt=ts sadpoint saddle
Frequency freq hessian∗ frequencies, freqPotential Energy Scan scan surface X
Excited State X X XReaction path following irc irc X
Molecular Dynamics admp, bomd drc dynamics, Car-ParrinelloPopulation Analysis pop pop X
Electrostatic Properties prop X XMolecular Mechanics X X X
Solvation Models X X X
X=⇒ method exists and keyword requires more than one options
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Job Types and Keywords II
Population Analysis in NWCHEM requires a PROPERTY · · · END block with variousoptions for different properties
propertydipolemulliken
end
Frequency calculations in GAMESS at end of optimization is carried out by addingHSSEND=.T. keyword in the STATPT control line
$STATPT HSSEND=.T. $END
Excited State Calculations include TDHF, TDDFT, CIS, CC methods
QM/MM Methods
Gaussian: ONIOM
GAMESS: Effective Fragment Potential, $EFRAG block
NWCHEM: task qmmm
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Job Types and Keywords III
Dynamics Calculations:
Gaussian:BOMD: Born-Oppenheimer Molecular DynamicsADMP: Atom centered Density Matrix Propagation (anextended Lagrangian Molecular Dynamics similar to CPMD)and ground state BOMD
GAMESS:DRC: Direct Dynamics, a classical trajectory method basedon "on-the-fly" ab-initio or semi-empirical potential energysurfaces
NWCHEM:Car-Parrinello: Car Parrinello Molecular Dynamics (CPMD)DIRDYVTST: Direct Dynamics Calculations usingPOLYRATE with electronic structure from NWCHEM
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Related HPC Tutorials
Fall Semester
Introduction to Gaussian/Electronic Structure Methods
Spring Semester
MD: Programming to ProductionApril 6th
Introduction to CPMD/Ab Initio Molecular DynamicsApril 27th
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Useful LinksAmber:http://ambermd.org
Desmond:http://www.deshawresearch.com/resources_desmond.html
DL_POLY:http://www.cse.scitech.ac.uk/ccg/software/DL_POLY
Gromacs:http://www.gromacs.org
LAMMPS:http://lammps.sandia.gov
NAMD:http://www.ks.uiuc.edu/Research/namd
CPMD: http://www.cpmd.org
GAMESS: http://www.msg.chem.iastate.edu/gamess
Gaussian: http://www.gaussian.com
NWCHEM: http://www.nwchem-sw.org
PINY_MD:http://homepages.nyu.edu/~mt33/PINY_MD/PINY.html
Basis Set: https://bse.pnl.gov/bse/portal
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Further ReadingDavid Sherill’s Notes at Ga Tech:http://vergil.chemistry.gatech.edu/notes/index.html
Mark Tuckerman’s Notes at NYU: http://www.nyu.edu/classes/tuckerman/quant.mech/index.html
Modern Quantum Chemistry: Introduction to Advanced ElectronicStructure Theory, A. Szabo and N. Ostlund
Introduction to Computational Chemistry, F. Jensen
Essentials of Cmputational Chemistry - Theories and Models, C. J.Cramer
Exploring Chemistry with Electronic Structure Methods, J. B. Foresmanand A. Frisch
Molecular Modeling - Principles and Applications, A. R. Leach
Computer Simulation of Liquids, M. P. Allen and D. J. Tildesley
♦ Modern Electronic Structure Theory, T. Helgaker, P. Jorgensen and J.Olsen (Highly advanced text, second quantization approach toelectronic structure theory)
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On LONI Linux Systems
/home/apacheco/CompChem
AIMD
CMD
ElecStr
ADMP
BOMD
CPMD
DRC
Amber
Desmond
Gromacs
LAMMPS
NAMD
OptFreq
PES
ExState
QMMM
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Using Gaussian on LONI Systems
Site specific license1 Gaussian 03 and 09
LSU Users: EricLatech Users: Painter, Bluedawg
2 Gaussian 03ULL Users: Oliver, ZekeTulane Users: Louie, DuckySouthern Users: Lacumba
3 UNO Users: No License
Add +gaussian-03/+gaussian-09 to your .soft file and resoftIf your institution has license to both G03 and G09, haveonly one active at a given time.
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Example Job submission script on Intel x86#!/bin/tcsh#PBS -A your_allocation# specify the allocation. Change it to your allocation#PBS -q checkpt# the queue to be used.#PBS -l nodes=1:ppn=4# Number of nodes and processors#PBS -l walltime=1:00:00# requested Wall-clock time.#PBS -o g03_output# name of the standard out file to be "output-file".#PBS -j oe# standard error output merge to the standard output file.#PBS -N g03test# name of the job (that will appear on executing the qstat command).
# setup g03 variablessource $g03root/g03/bsd/g03.loginset NPROCS=‘wc -l $PBS_NODEFILE |gawk ’//{print $1}’‘setenv GAUSS_SCRDIR /scratch/$USER# cd to the directory with Your input filecd ~apacheco/g03test# Change this line to reflect your input file and output fileg03 < g03job.inp > g03job.out
Linda Accessset NODELIST = ( -vv -nodelist ’"’ ‘cat $PBS_NODEFILE‘ ’"’ -mp 4)setenv GAUSS_LFLAGS " $NODELIST "g03l < g03job.inp > g03job.out
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Example Job submission script on P5#!/bin/tcsh# @ account_no = your_allocation# @ requirements = (Arch == "Power5")# @ environment = LL_JOB=TRUE ; MP_PULSE=1200# @ job_type = parallel# @ node_usage = shared# @ wall_clock_limit = 12:00:00# @ initialdir = /home/apacheco/g03test# @ class = checkpt# @ error = g03_$(jobid).err# @ queue
# setup g03 variablessource $g03root/g03/bsd/g03.login# setup and create Gaussian scratch directorysetenv GAUSS_SCRDIR /scratch/default/$USERmkdir -p $GAUSS_SCRDIR# cd to the directory with Your input filecd ~apacheco/g03test# Change this line to reflect your input file and output fileg03 < g03job.inp > g03job.out
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Sample Input%chk=h2o-opt-freq.chk%mem=512mb%NProcShared=4
#p b3lyp/6-31G opt freq
H2O OPT FREQ B3LYP
0 1OH 1 r1H 1 r1 2 a1
r1 1.05a1 104.5
Input Descriptioncheckpoint fileamount of memorynumber of smp processorsblank lineJob descriptionblank lineJob Titleblank lineCharge MultiplicityMolecule DescriptionZ-matrix formatwith variables
blank linevariable value
blank line
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Using GAMESS on LONI Systems
Add +gamess-12Jan2009R1-intel-11.1 (on Queenbee) toyour .soft and resoft
Job submission script#!/bin/bash#PBS -A your_allocation#PBS -q checkpt#PBS -l nodes=1:ppn=4#PBS -l walltime=00:10:00#PBS -j oe#PBS -N gamess-exam1
export WORKDIR=$PBS_O_WORKDIRexport NPROCS=‘wc -l $PBS_NODEFILE | gawk ’//{print $1}’‘export SCRDIR=/work/$USER/scrif [ ! -e $SCRDIR ]; then mkdir -p $SCRDIR; firm -f $SCRDIR/*
cd $WORKDIRrungms h2o-opt-freq 01 $NPROCS h2o-opt-freq.out $SCRDIRcp -p $SCRDIR/$OUTPUT $WORKDIR/
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Sample Input$CONTRL SCFTYP=RHF RUNTYP=OPTIMIZECOORD=ZMT NZVAR=0 $END
$STATPT OPTTOL=1.0E-5 HSSEND=.T. $END$BASIS GBASIS=N31 NGAUSS=6NDFUNC=1 NPFUNC=1 $END
$DATAH2O OPTCnv 2
OH 1 rOHH 1 rOH 2 aHOH
rOH=1.05aHOH=104.5$END
Input DescriptionJob control data
geometry search control6-31G** basis set
molecular data controlTitleSymmetry group and axis
molecule description inz-matrix
variables
end molecular data control
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Using NWCHEM on LONI Systems
Add +nwchem-5.1.1-intel-11.1-mvapich-1.1 (onQueenbee) to your .soft and resoft
Job submission script#!/bin/sh##PBS -q checkpt#PBS -M apacheco@cct.lsu.edu#PBS -l nodes=1:ppn=4#PBS -l walltime=0:30:00#PBS -V#PBS -o nwchem_h2o.out#PBS -e nwchem_h2o.err#PBS -N nwchem_h2o
export EXEC=nwchemexport EXEC_DIR=/usr/local/packages/nwchem-5.1-mvapich-1.0-intel-10.1/bin/LINUX64/export WORK_DIR=$PBS_O_WORKDIRexport NPROCS=‘wc -l $PBS_NODEFILE |gawk ’//{print $1}’‘
cd $WORK_DIRmpirun_rsh -machinefile $PBS_NODEFILE -np $NPROCS $EXEC_DIR/$EXEC \
$WORK_DIR/h2o-opt-freq.nw >& $WORK_DIR/h2o-opt-freq.nwo
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Sample Inputtitle "H2O"echocharge 0geometryzmatrixOH 1 r1H 1 r1 2 a1variablesr1 1.05a1 104.5endendbasis noprint
* library 6-31GenddftXC b3lypmult 1
endtask dft optimizetask dft energytask dft freq
Input DescriptionJob titleecho contents of input filecharge of moleculegeometry description inz-matrix format
variables used with values
end z-matrix blockend geometry blockbasis description
dft calculation options
job type
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