a brief introduction to gaussian

A Brief Introduction to Gaussian

CMBE

The users manual for gaussian is over pages long so a full discussionof the program is well beyond the scope of this document However this brieftutorial contains several examples and all the information you need to completeyour lab exercise

HO A Sample Run

A good way to start learning about gaussian is to try a small sample calculationCreate a le containing the lines below Note that the line numbers in the leftcolumn are just for reference and should not be typed Make sure you leave ablank line at the end of your le

HFSTOG TEST

Water Single Point Calculation

O

H O

H O H

The le you have just created is input deck for gaussian that will compute theenergy of a water molecule

Now run gaussian using the le you have created as input While you arewaiting for the run to nish read the next paragraph which explains what eachline in the input deck means

Line is called the route card and always starts with the sign in the rstcolumn Gaussian accepts input lines with a maximum of characters so ifyour route card is linger than characters you can continue it on subsequentlines but the sign is needed only on the rst line The route card ends witha blank line line in this case The format of the route card is always

methodbasis option option

where the slash is used to separate the method name from the basis setMultiple options can be used in the same run wih spaces to separate themA partial list of methods basis sets and keywords is given at the end of this

document Since Gaussian output can be very verbose it also recommendedto include the tag T immediately after the sign That is replace withT This switches on terse mode so that less information will be writtento the output le Terse mode will provide more than enough output detail forour purposes Line is the title card which gives a title and brief descriptionof the job It can run over one line if necessary Line terminates the titlecard Line gives the total charge and spin multiplicity of the molecule Thespin multiplicity is given by the expression s where s is the total spin ofthe molecule Note that paired electrons make no contribution to the spin sincethey have a net spin of zero Lines are called the Zmatrix The Zmatrixgives the positions of the atoms in terms of bond lengths and bond angles Wewill say more about the Zmatrix later

Analyzing the Output

A guassian ouput le gives considerably more data than we are interested infor this lab So rather than trying to explain everything in the output we willfocus on the portions that are most useful to us

Each time gaussian completes a selfconsistent evaluation of the energy itprints a report similar to this

SCF Done ERHF AU after cycles

Convg D VT

S

This tells us that selfconsistency has been achieved and the HartreeFock energy of the molecule is atomic units Since the goal of thiscalculation was to nd the energy of the molecule we are done

The Zmatrix

The Zmatrix is a method of representing atom positions using bond lengthsand bond angles The Zmatrix for water that we used earlier specied that thecalculation should contain an oxygen atom a hydrogen atom located Aaway and a second hydrogen also A away and forming a bond angle withthe rst hydrogen of degrees

To learn more about the Zmatrix lets construct the Zmatrix for hydrogenperoxide molecule HO whose structure is shown below

O

H

A A

AO

H

To construct the Zmatrix follow these steps

Step Choose an atom to represent the origin

The rst line of the Zmatrix consists solely of the atom label for thisatom The atom label is the atomic symbol with an optional number todistinguish atoms of the same type If we choose the left oxygen atom inthe illustration then the rst line of the Zmatrix is

O

Step Chose a second atom bonded to the rst and specify the bond lengthbetween the two atoms

The second line of the Zmatrix is the label for the second atom the labelfor the rst atom and the distance between them in A These items maybe separated wither by commas or whitespace If we use the left hydrogenas our second atom the Zmatrix becomes

O

H O

Step Choose a third atom bonded to either of the rst two and specify abond length and a bond angle

The next line of the Zmatrix will give the label of the third atom thelabel of an atom it is bonded to the bond length the label of a thirdatom and the bond angle formed by the three Our only choice in thiscase for the third atom is the second oxygen and the Zmatrix becomes

O

H O

O O H

Note that gaussian requires the bond angle to be in the range degrees It also must include a decimal point

Step Chose a fourth atom and specify a bond length bond angle and adihedral angle

The fourth line of our Zmatrix contains the label for the fourth atomthe label of an atom it is bonded to and the bond length the label of athird atom and the bond angle formed by the three and nally the labelof a fourth atom and the value of the dihedral angle formed by the fouratom chain

The dihedral angle describes the angle that the fourth atom makes withrepect to the plane formed by the other three It can range either from to degrees or from to degrees One way to visualize dihedralangles is called the Newman projection To form the Newman projection

imagine the bond between the two middle atoms of the chain is an axispoint straight out of the page Now draw the projections of the other twobonds as if they were the hands of a clock The angle measured clockwisefrom the atom being described between the hands is the dihedral angleThe Newman projection for hydrogen peroxide is

H

H

So the nal Zmatrix is

O

H O

O O H

H O O H

For molecules with more than four atoms Step is repeated until all theatoms are described

Constructing Zmatrices may seem dicult at rst especially if you arentused to thinking about dihedral angles However with a little practice you willnd it is much easier to translate your estimates for bond lengths and bondangles into a Zmatrix than it is to do the trigonometry to obtain cartesiancoordinates It is usually a good idea to use a program such as xmol which canread gaussian input decks to visualize molecules and check that your Zmatrixis correct

Optimizing Structures

So far we have specied all atomic coordinates exactly This is ne if we wantto calculate the energy a particular structure but it is not very useful if wewant to optimize the structure of a molecule In order to have guassian varythe atomic positions we need to introduce variables into the Zmatrix that tellgaussian which quantities can be changed

Lets perform a geometry optimization of hydrogen peroxide Create a lecontaining the following input deck

MPSTOG TEST OPT

Hydrogen Peroxide Geometry Optimization

O

H O ROH

O O ROO H A

H O ROH O A H D

ROH

ROO

A

D

Comparing this input deck to the earlier example we have changed from theHF method to the MP method We have also added the OPT keyword tothe route card to tell gaussian to perform a geometry optimization Observethat all of the distances and angles in the Zmatrix for HO we built earlierhave been replaced by variables and that initial guesses for all the variableshave been supplied in a new block that follows the Zmatrix Also notice thatwe have used the variables ROH and A twice to ensure that guassian preservesthe symmetry we expect the molecule to have Run gaussian with this inputdeck and ROH ROO A and D will be varied until guassian nds the minimumenergy structure

Analyzing the Output Part II

In performing a geometry optimization gaussian will perform several single pointcalculations as it varies the bond lengths and bond angles Gaussian will printthe results of each single point calculation just as in our water example In thiscase such calculations are performed as indicated by the lines

Search for a local minimum

Step number out of a maximum of

If we examine the energy in the last step we nd

SCF Done ERHF AU after cycles

Convg D VT

S

Since we are using the MollerPlesset correction the value of the correctionis given roughly lines after the HF energy on a line that looks like this

E D EUMP D

E is the value of the correction You can add the correction to the HF energyby hand or you can look in the archive block at the end of the le It looks likethis

Test job not archived

NATIONAL CENTER FOR SUPERCOMPUTING APPLICATIONSSIFFOptRMPFC

STOGHODRICHARDJul MPSTOG TEST OPTHydrogen Pe

roxide Geometry OptimizationO

HO

HVersi

onSGIGRevEStateAHFMPRMSD

eRMSFeDipolePGC XH

O

Its not very easy to read but most of the useful data from the run is in theresomewhere

Information about the optimized molecular structure can be found in severalplaces Several screens after the energy report you will nd a section like this

Optimization completed

Stationary point found

Optimized Parameters

Angstroms and Degrees

Name Definition Value Derivative Info

R R DEDX

R R DEDX

R R DEDX

A A DEDX

A A DEDX

D D DEDX

This shows the bond lengths and bond angles for our structure but its a littledicult to interpret since the variables names arent the same as the nameswe used in our input Zmatrix A few lines farther on you will nd a distancematrix which gives the distance between all the atoms in the molecule

Distance matrix angstroms

O

H

O

H

If you scroll a few more lines you will nd the cartesian coordinates of all theatoms

Standard orientation

Center Atomic Coordinates Angstroms

Number Number X Y Z

Finally in this case guassian has been able to gure out how to express thegeometry information in terms of our input Zmatrix Just above the archiveblock we see

Final structure in terms of initial Zmatrix

O

HROH

OROOA

HROHAD

Variables

ROH

ROO

A

D

Now we know the optimal structure and energy for hydrogen peroxide

Mulliken Population Analysis

By default Gaussian performs a Mulliken population analysis in which the totalcharge is partitioned among the atoms in the molecule For example a Mullikenanalysis on a formaldehyde molecule might give

Total atomic charges

C

O

H

H

Sum of Mulliken charges

Hence in this case a slight negative charge is placed on the Oxygen atomwith compensating positive charge distributed among the remaining atoms

It is important to note that a Mulliken population analysis is not a uniquescheme for assiging charge transfer since it is highly dependent on such factorsas the basis set

Molecular Orbitals and Orbital Energies

If the PopFull keyword is included in the route section then data about themolecular orbitals will be included in the output le They are printed at thebeginning of the population analysis section and for formaldehyde appear as

Molecular Orbital Coefficients

A O A O A O A O B O

EIGENVALUES

C S

S

PX

PY

PZ

O S

S

PX

PY

PZ

H S

H S

A O B O B O B V A V

EIGENVALUES

C S

S

PX

PY

PZ

O S

S

PX

PY

PZ

H S

H S

The atomic orbital contributions for each atom in the molecule are givenfor each molecular orbital with the MOs numbered in order of increasingenergyone MO per column Each MOs energy is listed in the row labeledEIGENVALUES Just above the eigenvalue row and below the MO number is arow giving the symmetry and occupancy occupied O unoccupied V of theMO The highest occupied molecular orbital HOMO and lowest unoccupiedmolecular orbital LUMO may be identied by nding the point where theoccupiedvirtual code letter in the symmetry designation changes from O to V

When determining an atomic orbitals contribution to a given MO the mostimportant thing is the relative magnitude of the atomic coecients regardlessof sign In the above example MO has several nonzero coecients butthe O s orbital coecient is more than an order of magnitude larger than theothers Hence this MO essentially corresponds to the O s orbital

Dummy Atoms

Sometimes you will nd that it is dicult to construct a Zmatrix with thedesired properties using only the atoms in the molecule Consider the case ofSiH Clearly all three HSiH bond angles should be identical You mightconsider a Zmatrix like this

Si

H Si Rhsi

H Si Rhsi H T

H Si Rhsi H T H D

Rsi

T

D

but this will not enforce the molecular symmetry In fact using this Zmatrixthe desired symmetry in only obtained when D and T are related in a specialway We can solve this problem by introducing a dummy atom X which doesnot interact but can be used to specify coordinates In this case we will use adummy atom as shown in the drawing below All of the XSiH bond angles canbe made identical thus ensuring the desired symmetry We obtain the Zmatrix

Si

X Si

H Si Rhsi X T

H Si Rhsi X T H

H Si Rhsi X T H

Rsi

T

Potential Energy Scans

The keyword SCAN in the route card instructs gaussian to scan the potentialenergy over the variables in the the Zmatrix For example you could scan thepotenial enegy of a hydrogen dimer using the following input

RHFSTOG TEST SCAN

Hydrogen Dimer Potential Scan

H

H H R

R

The last line of the le species the scan parameters In this case R will beginat A and scan additional points at A invervals The output of a scanwill include all the output for a single point calculation for each scan point anda summary of the energies at the end of the le For this case the summary is

Scan Completed

Summary of the potential surface scan

N R HF

Thermochemistry

Gaussian will compute the vibrational frequencies and other thermochemicaldata for optimized structures The easiest way to accomplish this is to use aroute card such as

MPG TEST OPT FREQ

which specifes both the OPT and the FREQ keywords Gaussian will rst optimize the structure and then compute vibrational frequencies using the structureit has just found Vibrational frequencies can be identied in the output bysearching for the word Frequencies The thermochemical data is written inthe following form

Temperature Kelvin Pressure Atm

Zeropoint correction HartreeParticle

Thermal correction to Energy

Thermal correction to Enthalpy

Thermal correction to Gibbs Free Energy

Sum of electronic and zeropoint Energies

Sum of electronic and thermal Energies

Sum of electronic and thermal Enthalpies

Sum of electronic and thermal Free Energies

If we are intereted in calculating the standard heat of reaction then the relevant energy is the one labeled Sum of electronic and thermal Enthalpies

Methods Basis Sets and Keywords

The following tables give a partial list of methods basis sets and keywordsknown to gaussian You should nd all that you need for your lab in thesetables

Basis Set Options

Basis Description

STOG Minimal basis set Use for more qualitative resultsG Double zeta Two sets of functions in the valence regionG Adds polarization functions to the heavy atoms These are

six component type d functionsG Adds polarization functions to hydrogens as well as heavy

atomsG Adds diuse functions Important for systems with lone

pairs anions excited statesG Adds p functions to hydrogens as wellG Triple zeta Adds extra valence functions sizes of s and

p functions to G These are ve pure d type functions

Method Options

Method Description

HF HartreeFock type calculation Unless specied RHF isused for singlets and UHF for higher multiplicites

MPn HartreeFock calculation followed by an nth order MollerPlesset correlation energy correction n

CIS HF calculation followed by conguation interaction with allsingle substitutions from the HF reference determinant

CISD HF calculation followed by conguation interaction with allsingle and double substitutions from the HF reference determinant

QCISD Quadratic CI calculation including sing and double substitutions

QCISDT Quadratic CI calculation including sing and double substitutions with triples contribution to the energy added

Keyword Options

Keyword Description

FORCE Perform a single analytical calulation of the forces on thenuclei

OPT Perform a geometry optimization over the variables listedin the variables block

FREQ Caclulate force constants virbational frequencies and otherthermochemical data

SCAN Scan the potential surfaceTEST Instructs gaussian not to archive the results of the calcula

tion in a central database

Quantum Mechanics Review Lecture 1 Page 1

Quantum Simulations ofMolecules and Materials

Many simulation methods have been developed based on quantum mechanicsto predict the molecular nature of molecules and materials. In this course we willfocus on:

•The application of these methods to solving current problems in researchHow to perform a simulationHow to choose a method of simulationHow to choose a basis sets, etc.How to understand and interpret our results

•We will briefly describe the fundamental theory behind these methods.

Examples of systems that can be simulated:

Reactions:•Docking of proteins, enzymes and substrates.•Reactions in combustion and atmospheric chemistry•Reactions on catalytic metal surfaces.•CVD reactions on surfaces for growing thin films.•Polymerization reactions•Diffusion through a material

Properties:•Molecular structure, including transition state structure•Energy, bond energies, activation energies, heats of reaction•Atomic forces•Vibrational frequencies, intensities, and transition moments•entropy and entropy of reaction•Dipole, quadrapole, etc. moments. Electron densities.•NMR constants•Force constants, •Potential energy surfaces, and kinetics


Reactor Modeling Scheme

ReactorReactants Products

Reaction Generators

Reactions andIntermediates

Kinetic Databases

QuantumChemistry

Thermodynamic and Kinetic Parameters

Reactive CFD

Agree with Experiment? No

Model Simplification?

Yes

Yes

Use Model No


Thermochemistry and Kinetics from Ab Initio

Enthalpy StructureEnergy

Zero-Point Energy

Vibrations

Entropy StructurePartition Function/

Energy Levels

Rotations

Vibrations

BarriersTransition State

StructureEnergy

Zero-Point Energy

Vibrations


Examples


Breakdown of Classical Physics

Although quantum mechanics has become an extremely useful method for predictingthe properties of molecules and materials, it was first developed to explain variouseffects which could not be explained by classical physics, including:

1) Black body radiation and the ultraviolet catastrophe:•Objects do not radiate energy at all frequencies, except at high T.•Electromagnetic oscillators are quantized and high energy modes are notactive unless enough energy is available.

2) The heat capacity of solids:•The heat capacity of solids does not follow the Dulong Petit Law at lowtemperatures.•The atoms of a solid vibrate together as quantized oscillators. Highenergy vibrations do not contribute to the heat capacity at low T.

3) The photoelectric effect:•Electrons are not ejected from a metal, no matter how intense the light ifthe frequency of the light is below some threshold.•The kinetic energy of the ejected electron increases linearly with thefrequency of the light, but does not depend on its intensity.•Light is composed of particles where the energy is associated with thefrequency and the number of particles with the intensity.

Quantum mechanics provided the solutions to the above quandaries by:

1) Quantizing the energies of a system.2) Combining the wave and particle-like nature of matter.

This was done primarily through the development of the wavefunction.


Postulates of Quantum Mechanics

Quantum mechanics is based on four basic postulates:

Postulate 1. The state of a system is fully described by the wavefunction.

Postulate 2. Observables are represented by operators which are linear, Hermitian and satisfy the following commutation relations:

Postulate 3. The mean value of the observable O in a series of measurements is equal to the expectation value of the corresponding operator. Measurement ofobservable O results in one of the eigenvalues of the operator O.

Postulate 4. The probability that a particle will be found in the volume elementdx at the point x is proportional to the square of the wavefunction times thevolume element dx.

[ ] ’’, qqq ipq δh= [ ] 0’, =qq [ ] 0, ’ =qq pp

Where q is a position operator (x,y,z) and pq is the corresponding momentumoperator.

Because the operators are Hermitian they have real eigenvalues and haveeigenvectors which form an orthonormal set.

measurement of position along an axis and momentum alongthat same axis don’t commute.


The Wavefunction ψ

The wavefunction plays a critical role in quantum mechanics. It is a mathematicalfunction:•whose square gives the probability of finding the particle in a given region of space•which contains all the measurable information about a system•whose curvature is proportional to the kinetic energy of the particle (the wavelengthof the particle is inversely proportional to the kinetic energy)•that is the solution to the Schrödinger equation

τψψ d*

1=ψψ

∫= τψψψψ d*

The probability of finding the particlein the volume dτ is:

The probability of finding the particlesomewhere is one:

The bra-ket notation:

The expectation value of the observable property O is:

∫== τψψψψ dOOO ˆˆ *

The operator correspondingto the observable O

The wavefunction

ψψ2

22

2 dx

d

MKE

h−=The kinetic energy is proportionalto the expectation value of the curvature:

)(2 xψ

dx

The wavefunction must besquare-integrable

ψ

x

ψ

ψ

ψ

ψ

x

xx

x


An important aspect of quantum mechanics is its ability to predict properties. Thisis done by finding a good wavefunction, and then finding expectation value of theproperty of interest. The following lists some common operators used in quantummechanics:

Position

Momentum

Kinetic Energy

Potential Energy

Total Energy

Angular Momentum

Operators

xipx ∂

∂−= hˆ

mzyxmK

22ˆ

22

2

2

2

2

2

22 ∇−=

∂∂+

∂∂+

∂∂−= hh

ˆ V = V (x, y,z)

ˆ H = ˆ K + ˆ V

∂∂−

∂∂−=

yz

zyiLx hˆ

ˆ X = x

∂∂−

∂∂−=

zx

xziLy hˆ

∂∂−

∂∂−=

xy

yxiLz hˆ


Superpostion principle

At any given instant a system may be in any eigenstate of a set of eigenstates fora given observable. If it is not in a single eigenstate, it is in a superposition ofeigenstates and it is not possible to predict the value of the observable. However, it is possible to predict the expectation value of the observable sinceit is just the mean of a large number of measurements of that observable sincethe squares of the amplitudes give the relative probabilities of measuring thepossible outcomes of the measurement.

Suppose the system is a superposition of states which are eigenstates of the operatorΩ: ∑=

nnnc ψψ

∫ ∑∑ Ω=Ω=Ω τψψψψ dccn

nnm

mmˆˆ **

The expectation value of Ω is:

∫∑ Ω=Ω τψψ dcc nmnm

nmˆ*

,

*

and

Since the eigenvectors of Ω form an orthonormal set, the integral abovevanishes except when m and n are equal:

∑∑∫∑ ===Ωn

nnn

nnnnnn

nnn cccdcc ωωτψψω 2***

nnn ψωψ =Ω

Thus the expectation value of the observable Ω is the weighted sum of theprobabilities of finding each particular eigenstate times the eigenvalue of thateigenstate (this might remind you of statistical mechanics).

Measuring Ω leaves the system in a definite state: that is the act of observationforces the system into an eigenstate of that observable, although which eigenstate we find completely depends on the initial state: the relative weightingsof the different eigenstates in initial superposition state.


The Schrödinger Equation

The wavefunction of a system is the one that solves the Schrödinger equation (thiscan be taken as a postulate of quantum mechanics). The Schrödinger equation is aneigenvalue equation which has the wavefunction as its eigenvector , the Hamiltonianas its operator and the energy as its eigenvalue.

ˆ H ψ = Eψ

EH =ψψ ˆThe expectation value of the Hamiltonian is the total energy E:

For the molecular systems we will consider the Hamiltonian will include the potential energy due to the Coulombic interactions in the molecule:

∑∑∑>>

++−==ABA AB

BA

iji ijAi iA

A

R

eZZ

r

e

r

eZzyxVV

,

2

,

2

,

2

),,(ˆ

∑∑∑∑∑>>

++−∇−∇−=ABA AB

BA

iji ijAi iA

A

i A

A

i

i

R

eZZ

r

e

r

eZ

MmH

,

2

,

2

,

22222

22ˆ hh

Where the molecular Hamiltonian is then:

Note that the Hamiltonian depends on the wavefunction, and that thewavefunction depends on the Hamiltonian. This makes finding the wavefunctiondifficult! However, all procedures for finding the wavefunction depend on finding approximate solutions to the Schrödinger equation.

A

B

i j


Many-particle wavefunctions

The Pauli Exclusion Principle dictates that up to two electrons can sharethe same spatial distribution (orbital) and if two electrons are in the sameorbital they must have opposite spin.

The above requirement is not automatically satisfied by solutions of theSchrödinger equation because the Hamiltonian does not explicitly depend on spin. However, this requirement can be satisfied by imposing the condition that the wavefunction must be antisymmetric under electron interchange.

So an antisymmetric wavefunction will change sign if the coordinatesof any two electrons are exchanged:

),...,,...,,...,(),...,,...,,...,( 11 nrsnsr xxxxxxxx ψψ −=

The requirement that the wavefunction be antisymmetric is called theAntisymmetry Principle and is a general statement of the Pauli ExclusionPrinciple.

Later we will see that this requirement can be easily imposed by formingwavefunctions in the form of Slater Determinants.


Electronic Hamiltonian and Wavefunctions

The wavefunction must be antisymmetric with respect to the electronic coordinates, but what about the nuclear coordinates?

Since the nuclei are much heavier than the electrons the nuclear kinetic energyis neglected and nucleus-nucleus interactions are considered parametrically:

∑∑∑>

+−∇−=iji ijAi iA

A

i

ielec r

e

r

eZ

mH

,

2

,

222

2ˆ h

∑∑∑∑∑>>

++−∇−∇−=ABA AB

BA

iji ijAi iA

A

i A

A

i

i

R

eZZ

r

e

r

eZ

MmH

,

2

,

2

,

22222

22ˆ hh

Small Constant

This results in a Hamiltonian called the Electronic Hamiltonian

A

B

i j

Here the electronic wavefunction is solved for a given set of atomiccoordinates. The nuclear coordinates are not explicitly stated in thewavefunction usually.

),...,,( 21 nxxxψ

( ) ( )NBAxxxNBAxxx nNne ,...,,),...,,(,...,,),...,,( 2121 δψψψ ==Ψ

Since the nuclei don’t move on the same time scale as the electrons (their motions are decoupled), the nuclear wavefunctions look likedelta functions on the electronic time scale:


Spin and Spatial Orbitals

For two electrons that have the same spatial quantum numbers,the spatial distribution of probability amplitude is the same and is called an orbital or spatial orbital.

1s 2p 3dVarious AtomicOrbitals:

If two electrons occupy the same spatial orbital, they must havedifferent spin.

The spin orbital is the product of the spatial orbital and the spinfunctions.

( ) ( )ωβωαψχ ),(),( 2121 xxxx =

Spin Orbital

SpatialOrbital

SpinFunctions

ββααψψψαβψαβχχ ==1== ββαα

0=βα


Hartree-Products

What form must the wavefunction take for a many-body system?

Since the wavefunction is a probability amplitude its square must have the properties of a probability function:

P(x1, x2) = P(x1)P(x2)

If electron 1 and 2 are independent, the probability of finding electron 1 with coordinates x1 and electron 2 with coordinates x2 must equal the product of the independent probabilities of finding electron 1 and electron 2 with coordinates x1 and x2.

ψ (x1, x2) = ψ (x1)ψ (x2 )...

)()()()(),(),( 21212121 xxxxxxxx ψψψψψψ =

)()()()(),(),( 22112121 xxxxxxxx ψψψψψψ =

and the two-particle probability density is:

Where we remove the constant terms from the respective integrals:

This is of the form:)()(),( 2121 xPxPxxP =

Hartree-Product

So the n-particle wavefunction must have the form:


Slater Determinants

Although the Hartree Product has the correct form for the probability density, it is not antisymmetric:

)()()()( 12212211 xxxx ψψψψ −≠

The Slater determinant is a way of ensuring both the antisymmetry of thewavefunction, as well as the correct form of the probability:

( )

)()()(

)()()(

)()()(

,...,2,1

21

22221

11211

nnnn

n

n

xxx

xxx

xxx

n

φφφ

φφφφφφ

ψ

L

MOMM

K

K

=

Each column is adifferent spin orbital

Each row is adifferent electron

If any two rows are exchanged, the wavefunction changes sign: The wave-function is antisymmetric.

The form of the wavefunction is a sum of Hartree-products, each with asign depending on whether the permutation is even or odd.

This also allows for correctly treating the indistinguishability of the electrons. (Each electron finds itself in every orbital with equal weight).

Quantum Chemistry Basics Lecture 4 Page 1

The Variational Principle

Since according to the variational principle the expectation value of H for any approximate wavefunction is always larger than that of the exact ground state:

0EH ≥

∑=i

iitrial c ϕψ ∑∑=i

iii

ii cHcH ϕϕ ˆ

A method for detemining wavefunctions can be developed which uses the expectation value of H as measure of the quality of the wavefunction: That is we want to minimize the quantity:

tt HH ψψ ˆ=

When we proved the variational principle we used a basis of the eigenfunctions of H. Here we cannot do that because the wavefunction is the unknown. In fact, we don’t even know the Hamiltonian:

If we expand the trial wavefunction in a set of basis functions then we have variationalparameters: the coefficients of the basis functions.

∑∑∑>

+−∇−=iji ijAi iA

A

i

i

r

e

r

eZ

mH

,

2

,

222

2ˆ h

A

B

i j

Electron-Electroninteraction depends on the wavefunction

Electron-nuclearinteraction depends on the wavefunction

These shouldbe electrondensities


The Hartree-Fock Approximation

Although we can use the variational principle to solve the Schrodinger equation, the method ofsolution is not obvious because the wavefunction depends on the Hamiltonian, and the Hamiltoniandepends on the wavefunction:

( )ψH ( ) ( ) )ˆ()ˆ(ˆ HEHH ψψψψ =

Hartree and Fock independently developed a method to overcome this difficulty:

1: Guess a trial wavefunction (this amounts to guessing the basis coefficients)2: Calculate the expectation value of F for this trial wavefunction where

F is an operator which replaces the electron-electron Coulomb interactionwith the interaction of each electron with the average field of the other electrons.

3: Minimize the expectation value of F with respect to the expansion coefficients4: Take the new wavefunction from step 3 and repeat steps 2 and 3 until he new

wavefunction obtained is equal to the one from the previous iteration.

So instead of solving the Schrodinger equation we solve the Hartree-Fock equations:

( ) )()(ˆ iiiF χεχ =

)(2

ˆ,

2

ir

ZF HF

Ai iA

A

i

i ν+−∇−= ∑∑


The Hartree-Fock Method

•Because HF is an interative procedure which ends when self-consistency is reached(according to some threshold criteria) it is called a Self-Consistent Field theory.

•The word field signifies the property that electrons only interact in an average way:they move in the average field of the other electrons.

•HF does not contain dynamic correlation: That is the only correlation in HF is that dueto the average field, and that due to exchange correlation between like spins.

•The improvements more accurate methods based on HF make are due to includingdynamic correlation: usually just called correlation.

•HF is one of the most efficient ab initio methods: Its characteristics are:•It overestimates vibrational frequencies by about 10 percent.•It produces very accurate geometries•Transition states energies are usually overestimated (50-300 percent)•Bond energies are underestimated.

HF is a good base theory, and often used for initial exploration of the properties of asystem.


Slater Determinants and Probabilities

For the doubly occupied orbital:

( ) αβϕϕβαϕϕαβϕϕβαϕϕψ ssssssss 11111111

22,1 −−=

Note that the cross terms vanish due to the orthogonality of the spin functions:

( ) αϕαϕβϕβϕβϕαϕαϕβϕαϕβϕβϕαϕβϕβϕαϕαϕψ ssssssssssssssss 1111111111111111

22,1 +−−=

Electron 1in orbital 1,Electron 2in orbital 2




Integral overcoordinates ofelectron 1

Integral overcoordinates ofelectron 2

( )111111211111

22,1 ssssssss ϕϕϕϕϕϕϕϕψ +=

Also note that this wavefunction was not normalized: When making Hartree-product, orSlater determinant wavefunctions we add a normalization factor:

( )

)()()(

)()()(

)()()(

1,...,2,1

21

22221

11211

nnnn

n

n

xxx

xxx

xxx

nn

φφφ

φφφφφφ

ψ

L

MOMM

K

K

=

There is NO correlation between the motions of electrons with opposite spin.


Probabilities and Correlation

For the doubly occupied orbital:

( ) ααϕϕααϕϕααϕϕααϕϕψ ssssssss 12211221

22,1 −−=

Integrating over the spin coordinates:

( ) αϕαϕαϕαϕαϕαϕαϕαϕαϕαϕαϕαϕαϕαϕαϕαϕψ ssssssssssssssss 1122211212212211

22,1 +−−=





( ) ssssssssssssssss 1122211212212211

22,1 ϕϕϕϕϕϕϕϕϕϕϕϕϕϕϕϕψ +−−=

The cross-term does not vanish: There is Correlation between the motions of electronswith the same spin: This is called Exchange Correlation. Hartree Fock contains this Correlation automatically.


Basis Functions

There are many choices in making an expansion of atomic-like wavefunctions in a set of basis functions:

∑=

=k

iiic

1

ϕψ

We can include basis functions with the same nodal behavior as the atom: that is with the sameangular momentum quantum numbers:

H: 1s functionsC: 1s, 2s and 2p functionsSi: 1s, 2s, 2p, 3s, and 3p functions

To improve the description of bonds we often include polarization functions: this improvesthe ability of the basis set to describe distortion of electrons. We might add:

H: 2p functions (and even 3d functions if very high accuracy is desired)C: 3d functions (and even 4f functions)Si: 3d functions (and even 4f functions), etc.

Basis withtight, mediumand diffusefunctions.


Basis Functions

Diffuse functions can also be added: These functions have the same angular momentum quantum numberas the valence shell, but are even more diffuse than those included in double zeta, and triple zeta basis sets:

Double zeta refers to the number of different exponents used to describe a valence orbital. Thus,double zeta means using tight and diffuse functions. Triple zeta means using tight, medium andloose functions.

∑=

=k

iiic

1

ϕψ ( )2exp ri ζϕ −=( )ri ζϕ −= expThe function becomes tighter as zeta becomes larger.

Slater-like (STO) Gaussian

Gaussian functions are generally used because integrals of Gaussian functions, and products of Gaussianfunctions are simpler than integrals of Slater functions. However, this introduces an error because theCusp at r=0 is not sharp for a Gaussian function unless zeta goes to infinity: To overcome this Contractionsare used:

A contraction is a fixed sum of Gaussian functions of varying zeta which approximate the Slater form andtogether are taken as a single basis function:

This is especially important for core orbitals, but is often included for tight valence orbitals as well:

∑=

=m

jjji g

1

ηϕ


Basis Sets

3-21G One triply contracted core function, one doubly contracted tight valence function, a loose valence function 6-31G One six-fold contracted core function, one triply contracted tight valence function, a loose valence function3-21G(d) One triply contracted core function, one doubly contracted tight valence function, a loose valence function

One set of d polarization functions on heavy atoms6-31G(d) One six-fold contracted core function, one triply contracted tight valence function, a loose valence function

One set of d polarization functions on heavy atoms6-31G(d,p) One six-fold contracted core function, one triply contracted tight valence function, a loose valence function

One set of d polarization functions on heavy atoms, a set of p polarization functions on H6-311G(d) One six-fold contracted core function, one triply contracted tight valence function, one medium and one

loose valence function. One set of d polarization functions on heavy atoms6-311G(d,p) One six-fold contracted core function, one triply contracted tight valence function, one medium and one

loose valence function. One set of d polarization functions on heavy atoms, and a set of p polarization on H.6-311G(2df,p) One six-fold contracted core function, one triply contracted tight valence function, one

medium and one loose valence function. A tight and loose set of d polarization functions on heavy atoms, a set of p polarization functions on H.

6-311+G 6-311G plus a set of diffuse valence functions on heavy atoms6-311++G 6-311G plus a set of diffuse valence functions on heavy atoms, and H.

6-311++G(3df,pd)

Quantum Chemistry HF Basics Lecture 5 Page 1

Basis Sets

6-311++G(3df,pd)

6-fold ContractedCore

Triple Valence3-fold tightValenceContraction

Diffuse Functions on Heavy andLight Atoms

3 d Sets and an F Set of Polarizationon Heavy Atoms

p Set and an d Set of Polarizationon H

6-31+G(2d, p)

6-fold ContractedCore

Double Valence3-fold tightValenceContraction

Diffuse Functions on Heavy Atoms

2 d Sets of Polarizationon Heavy Atoms

p Set of Polarizationon H


The Variational Principle

The variational principle states that the expectation value of H for any approximate wavefunction is alwayslarger than that of the exact ground state:

0EH ≥

∑=i

iitrial c ϕψ

∑∑ −=−i

iii

ii cEHcEH ϕϕ 00ˆ

∑∑ −=−i

iiii

ii cEEcEH ϕϕ 00 ’

( ) ’’ 0’*

0 iii

iii EEccEH ϕϕ∑ −=−

( ) 0’ 0*

0 ≥−=− ∑i

iii EEccEH

Consider a trial wavefunction expandedin a basis of eigenfunctions of H

We can write:

Substitute in our expansion:

tit EHEH ψψ 00ˆ −=−

Replace the operator H by its eigenvalues:

Remove constants from the integral:

Include the orthogonality and signof the integral:


Rayleigh-Ritz Method

The Rayleigh-Ritz Method uses a linear combination of fixed basis functions to apply the variational principle.

∑=i

iitrial c ϕψ

ijji

ji

ijji

ji

jiji

ji

jiji

ji

jjj

iii

jjj

iii

Scc

Hcc

cc

Hcc

cc

cHc

H∑∑

∑∑

∑∑

∑∑====

,

,

,

,

ˆˆ

ˆϕϕ

ϕϕ

ϕϕ

ϕϕε

Consider a trial wavefunction expandedin a fixed basis

We can write:

Substitute in our expansion:

tt

tit HH

ψψψψ

εˆ

ˆ ==

To apply the variational principle wedifferentiate with repect to each ck:

This is satisfied when the seculardeterminant is zero:

( ) ( )0

,,,

, =

−+

−=

=

∑∑

∑∑

∑∑

ijji

ji

ikiki

i

ijji

ji

kjkjj

j

ijji

ji

ijji

ji

kk Scc

SHc

Scc

SHc

Scc

Hcc

dc

d

dc

dεε

ε

0=− ijij SH ε


The Hartree-Fock Approach

The spinorbitals that give the lowest energy for a single determinant wavefunction are found using the variationalMethod.

The Hartree-Fock approach involves replacing the full Hamiltonian with a sum of one electron Hamiltonians where each electron interacts with the average field of the other electrons.

)()(1 iif χεχ =

)(2

ˆ,

2

ir

ZF HF

Ai iA

A

i

i ν+−∇−= ∑∑ )1(2 1

2

1HF

A A

Ai

r

Zf ν+−∇−= ∑

)1()1()1(1 uu

u KJhf ∑ −+=

)1()2(1

)2()1()1(12

auuau rJ φφφφ

=

)1()2(1

)2()1()1(12

uauau rK φφφφ

=

Coulomb Operator:

Exchange Operator:

N Fock equations:


Hartree-Fock Method

Choose Basis Functions

Choose Coefficients

(Starting Guess)

CalculateF

Calculatedet |F-εS|

CalculateOverlap matrix S

CalculateEnergies and

New coefficientsConvergence

yes

no

Choose Structure,Spin and Charge


Configuration Interaction

The HF wavefunction is the lowest energy single determinant solution to the Schrodinger equation:

Other wavefunctions based on HF can be written which have multiple determinants (configurations):

∑∑∑ +++=t

ttd

dds

ss cccHFCI

ψψψψψ

)()()(

)()()(

)()()(

1

2221

1111

nnnrn

nr

nr

r

xxx

xxx

xxx

φφφ

φφφφφφ

ψ

L

MOMM

K

K

=

One HF spin orbital replaced by a HFvirtual orbitalSingly excited determinant:

A doubly excited determinant would have two HF spin orbitals replacedby virtual orbitals

The CI wavefunction is optimized by variationally determining the CI coefficients.

Quantum Chemistry CI Basics Lecture 6 Page 1

Contraction Examples

∑=

=k

iiic

1

ϕψ ∑=

=m

jjji g

1

ηϕ

Spatial parts of spin orbitals are expanded over a set of basis functions:

Since the actual spin orbitals are slater like,contractions of Gaussian functions are usedto approximate the cusp of a Slater Type Orbital.

r

r

∑=

=m

jjji g

1

ηϕ

cuspψ

3-21G 6-31G 6-31G(d) 6-31G(d,p) 6-311G(d,p) 6-311+G(d,p) 6-311+G(2df) 6-311++G(2df,p)H 2 2 2 5 6 6 3 71st Row 9 9 14 14 18 22 37 372nd Row 13 13 18 18 22 26 41 41

1s 2s 2p 3s 3p 3d6-31G(d) H 2 0 0 0 0 0

C 1 2 6 0 0 5Si 1(6) 1(6) 3(6) 2 6 5

6-311+(d,p) H 4 0 3 0 0 0C 1(6) 4 12 0 0 5Si 1(6) 1(6) 3(6) 4 12 5


Single Excitations

The HF wavefunction is the lowest energy single determinant solution to the Schrodinger equation. Using theHF wavefunction as a reference we form singly excited determinants by replacing each orbital by a virtualorbital.

)()()(

)()()(

)()()(

)()()(

1

3331

2221

1111

nnnrn

nr

nr

nr

ra

xxx

xxx

xxx

xxx

φφφ

φφφφφφφφφ

ψ

KK

MOMMM

KK

KK

KK

=

One HF spin orbitalreplaced by a HFvirtual orbital

)()()(

)()()(

)()()(

)()()(

1

3331

2221

1111

nnnan

na

na

na

xxx

xxx

xxx

xxx

φφφ

φφφφφφφφφ

ψ

KK

MOMMM

KK

KK

KK

=

Brillouin’s Theorem: Singly excited determinants do not directly mix with the ground state.

HFHFCISs

ssc ψψψψ =+= ∑

HFHFHFHFHFCISCISHHccHcHH r

arass

ras ψψψψψψψψψψ =++= ∑∑


Doubly Excited Determinants

Using the HF wavefunction as a reference we form doubly excited determinants by replacing two orbitals by two virtual orbitals.

)()()()(

)()()()(

)()()()(

)()()()(

1

33331

22221

11111

nnnsnrn

nsr

nsr

nsr

rsab

xxxx

xxxx

xxxx

xxxx

φφφφ

φφφφφφφφφφφφ

ψ

KKK

MOMMM

KKK

KKK

KKK

=

The HF ground state can directly mix with doubly excited determinants: Thus, because of Brillouin’s theorem we expect that double excitations will be the leading corrections to the HF wavefunction.

∑+=d

rsabdc

HFCIDψψψ

∑∑ ++= rsab

rsabdd

rsabd HccHcHH

HFHFHFCIDCIDψψψψψψψψ

Higher and higher order configurations can be used to improve the wavefunction, however thecorrections get smaller and the expense of the calculation grows geometrically.

HF

Singles Triples

Doubles


CI


Choose Coefficients

(Starting Guess)

CalculateF





yes

no


ChooseCI expansion

Variationally solve forCI expansioncoefficients

HF Loop


Multi Reference CI

In addition to expanding the wavefunction based on configurations made from exciting out of a HF reference, we can also use multiple reference states in the CI expansion. This might be done if certainelectronic configurations are expected to be important:

...’... ++++++= ∑∑∑∑D

drefd

S

Srefs

D

dHFd

S

SHFs cccc

HFMRCIψψψψψψ

The reference states are often chosen as low lying excited states which have replace orbitalsthat are active in the process being considered (say a chemical reaction).

HF

Singles Triples

Doubles

S T

D


MCSCF


Choose Coefficients

(Starting Guess)

CalculateF





yes

no


ChooseCI expansion

Variationally solve forCI expansioncoefficients

HF LoopHold CI expansion

fixed and reoptimizeorbitals

MCSCF Loop


Multiconfiguration SCF

MCSCF involves first doing a CI calculation using the fixed spin orbitals (usually from HF) to determine the CIexpansion coefficients, then reoptimizing the orbitals and repeating the process iteratively until self-consistency is reached.

...+++= ∑∑D

dHFd

S

SHFs cc

HFMRCIψψψψ

This process thus involves an inner and outer iterative process to obtain the best wavefunction.

NOTE: MCSCF wavefunctions are extremely expensive to calculate! Many scale as N6.

Examples of MCSCF methods include: CASSCF, and GVB*CI.

HF

Singles Triples

Doubles

QC Practical Issues Lecture 7 Page 1

Symmetry

Molecular symmetry can be used to reduce the cost of an electronic structure calculation:

• Linear combinations of atomic orbitals can be constructed which are representations of the overall molecular symmetry. These are called symmetry adapted linear combinations of AO.

•Molecular orbitals can be formed by combining these linear combinations of atomic orbitals.

• The symmetry of the underlying SALC imposes relationships between the two electron integrals between electrons.

These relationships can be used to determine integrals without calculating them.


Symmetry Adapted Linear Combinations

Combining atomic orbitals of atoms that are related by symmetry:

3 Nodes

0 Nodes

1 Node

2 Nodes


Molecular Orbitals from SALC of Atomic Orbitals

We can make combinations of SALC of atomic orbitals to form molecular orbitals:

+

+ +

+/-

- +


Symmetry

Note: Not all symmetry can be used by QC methods (only abelian groups can be used).

If mirror planes and Cn axis exist in your system, input molecules such that this symmetry exists and can berecognized by the program.

•Use geometry variables that are the same for distances, angles etc. that are related by symmetry.•Use high symmetry planes, axis to define your molecule.

Speed improvements are approximately a factor of 2 for each used mirror plane.

Interpretation of nodal planes, molecular orbitals etc. will be easier if symmetry is used.

C3

C2

Note: Program can take Advantage of the C3 axis:

In order for program to use mirror plane, dihedral anglemust be 180 degrees andsymmetry related bonds and anglesmust be input equally.


Bond Energy

Bond energy is the difference between the energy of a molecule and two fragments of the moleculemade by separating an atom from the rest of the molecule.

•This can be done by calculating the fragments separately, or together (spaced far apart).Because of scaling not being linear, better to calculate separately (usually).

•If the molecule is not allowed to relax, then the bond energy is called a snap bond energy.

Optimize structure of CH3 fragment


Charge State and Spin State

When inputting a QC calculation the charge state of the molecule must be specified. The charge statemust be an integer. Specifying charge states other than zero allows the calculation of ions.

To calculate the electron affinity of a molecule calculate the energy difference betweenthe molecule and the anion state (charge state -1)

To calculate an ionization potential, calculate the energy difference between a molecule and itscation state (charge state +1). Second and third ionization potentials can be calculated in a similar fashion.

The spin state of the molecule must also be specified when inputting an electronic structurecalculation.

•For closed shell systems, the spin state is a singlet (s=0)•For a system with a single dangling bond, the charge state is usually a doublet (s=1/2)•For a system with multiple unpaired electrons, the charge state is not oftenobvious.


Ethylene example

Ethylene: Closed shell (singlet), neutral molecule.4 CH bonds, a C-C sigma bond, and a C-C pi bond.CH ~ 1.09Å, CC ~ 1.31 Å. 3 mirror planes.

C-C pi bond

C-C sigma bond

C-H bonds

C 1s Core

Energy

C-C pi antibond

C-C sigma antibond

To calculate EA: change charge state to -1and spin state to 1. Recalculate energy (opt?)

To calculate IP: change charge state to +1and spin state to 1. Recalculate energy (opt?)

Calculating triplet state involves changingspin state to 2 (opt?). An electron is promotedto the pi antibond.

HOMO: The highest occupied molecularorbital.

LUMO: The lowest unoccupied molecularorbital.

HOMO

LUMO


Methylene example

Methylene: Neutral molecule. Singlet or triplet?2 CH bondsCH ~ 1.09Å2 mirror planes. (C2V symmetry, like water).

C-asymmetric p pair

C-symmetric p pair

C-H bonds

C 1s Core

Energy

C-asymmetric p pair

C-symmetric p pair

C-H bonds

C 1s Core

Energy

QC Optimizations and Correlation Lecture 8 Page 1

Ethylene example

Ethylene: Closed shell (singlet), neutral molecule.4 CH bonds, a C-C sigma bond, and a C-C pi bond.

Enthylene dissociates into two triplet CH2 methylenes.

To calculate the dissociation energy the fragments mustbe in their triplet state.A better dissociation energy wouldbe found by allowing the CH2 groups to relax.

The rotational barrier is overestimated if the molecule isnot allowed to relax. This is because the CC distance istoo short for a sigma bond. The molecule compromises in the case where it can form a pi bond, but when the pibond is broken the molecule will relax to its equilibriumC-C single bond distance (about 1.5Å).

E

RCC

E

RCC


Methylene example

Methylene: Neutral molecule ground state is a triplet!(while SiH2 is a singlet).

sp3

sp2

sp3

sp3

sp2

In order to improve the bonds, C increases the S character leaving a dangling bond a pure p state. Sinceit costs energy to (hybridize) increase the p character a lower energy can be obtained in the sp2 state. Also note that for C less p character increases the HCH angles moving the Hs further apart from each other.

For silicon the p orbitals are relatively small and the hybridization energy is low as well. It thus takes morep character to make a good bond and little penalty must be paid in order to hybridize.


Correlation Energy

The correlation energy of a molecule is defined as the difference between the exact energy and the HFenergy. HF doesn’t allow electrons to instantaneously interact and thus dynamic correlation is not included.

In order to describe many effects it is crucial to include dynamic correlation. Thus many methods have beendeveloped in order to improve upon the HF wave function (CI methods, for example).

ψ g = ψ l + ψ rψ u = ψ l − ψ r

ψ = ψ g +ψ u ψ = ψ g −ψ u

Using the antibond configuration in a CI expansion allows the electronic wavefunction to correlatethe electron motion between the left and right H atoms. This becomes more important as the moleculedissociates.


Correlation Energy and Bond Strengths

The most important kind of correlation for describing bonding and transition states is ‘left-right’ correlation.•Bonds

The next most important type of correlation is ‘angular correlation’. To describe angular correlation it isimportant to include polarization functions in the basis set expansion.

•Lone pair electrons.

In-out correlation is radial correlation. Here an electron near a nucleus will cause other electrons to prefer larger distances from that nucleus.

Hartree-Fock underestimates bond strengths because it does not include dynamic electron correlation.

When a molecule dissociates the electrons are found on their respective nuclei: there is nearly completecorrelation between the electron motion.

HF on the other hand forces the electrons to both be in the vicinity of one nucleus with independent probabilities, even when there is infinite separation between the nuclei. The energy of the dissociated state is severely overestimated.

ψ g = ψ l + ψ rψ g = ψ l + ψ r


Geometry Optimizations

One of the primary goals of QC is to predict atomic structures, including transition states.•In many cases the structure itself is the quantity of interest.•In all cases the energy depends on the structure so obtaining good energies meansusing good structures.•Examples of this include the improvements that can be made from snap bonds energies,rotational barriers, and other energies when the molecule is allowed to relax during a distortion.

Question: How do you predict new structures?

Short answer: Use the optimization (opt) or transition state (TS) keywords in Gaussian…

Question: What is Gaussian doing?

Answer: It uses the Hellmann-Feynman Theorem to calculated forces.

dEdP

= d ˆ H dP

The derivative of the energy with respect to some parameter is the expectation value of the derivative ofthe Hamiltonian with respect to that parameter.

Since forces are derivatives of energy with respect to moving a nucleus we must determine derivativesof H with respect to the nuclear coordinates.


Geometry Optimizations

Once the wave function is obtained for a given structure, forces at that geometry are obtained bytaking the expectation values of the derivatives of the Hamiltonian for that wave function:

dE

dRA= d ˆ H

dRA= ψ d ˆ H

dRAψ

Once the forces are obtained, the atoms are allowed to move according to the forces, the masses of the atoms and a quasi-time that the forces are allowed to act. Gaussian chooses the step size, but it canbe changed.

The nuclei thus move in the potential described by the energy as a function of geometry.

Once the new structure is obtained the wave function has to be calculated at that structure: the derivativesevaluated and new forces calculated. These forces are allowed to act to produce a new structure and theprocess repeats until:

•small RMS forces are obtained.•No one force exceeds some maximum value•small RMS displacements are obtained.•No one displacement exceeds some maximum value.


Optimizations

Although structure prediction can be very accurate, several key points should be kept in mind whenusing Gaussian to predict new structures:

•Use as good of a guess for a structure as you can: You can save enormous amounts ofCPU time.

•Structures are not generally very sensitive to basis sets: you can use a more approximatebasis set to do the first rough optimization and refine the optimization later with a bigger basis.•HF generally predicts structure quite accurately.

•If you have no idea what structure to use, find molecules with analogous fragments/groupsand use those structural parameters as starting guesses.

•Be careful not to allow symmetry to be too high: Too much symmetry will cause some forcesto be zero: example: planar ammonia.

•If you don’t know your molecule/system well and the system is quite large, optimizations offragments of the molecule can help for starting points for determining the structure of thewhole system.

•If atoms are equivalent, use equivalent parameters to describe the atomic positions which will reduce the number of degrees of freedom and speed up the calculation.

For difficult cases force constants can be used (the book describes these options on pages 47-48)


Optimizations

Be aware that an optimization will tend to find the nearest local minimum of the energy surface.

Thus, the predicted geometry may not be the correct one: again, the starting guess is important.

It may also be necessary to check different conformers, etc.

Optimizations are much slower than single point calculations:

Expect calculations to take longer!


Transition States

E

RHH

Transition state: often, as a molecular systemundergoes a chemical transformation it mustgo over an activation barrier.

The energy is the energy of a saddle point on the potential energy surface for the chemical reaction.

This usually occurs because a system must first partially breakexisting bonds in order to form new bonds.

Example: H2 + H

As an H atom approaches H2 molecule, the good overlap withthe closest H atom in H2 is not enough to make up for the perturbationthis electron has on the original H2 bond. (they mustbe orthogonal to each other).

0 Nodes SALC

1 Node SALC

2 Nodes SALC

I must put 3 electrons into the threeorbitals at the left. Two in the lowestenergy orbital and one in the secondlowest energy orbital (which is non-bonding).


Transition States

Gaussian can optimize for transition states by minimizing the forces on the molecule with therestriction that the molecule have one negative frequency (the molecule is at a saddle point).

Notes: •Transition state searches are sometimes difficult because the wave function is harderto describe at the TS.•The transition state geometry is harder to guess. Often times we guess structures where theactive bonds are stretched 15 to 20 percent.•TS are usually very sensitive to basis sets, and level of theory.

To optimize to a TS use one of the TS keywords: •Some key words use interpolation between reactants and products to guess structures.•Other methods use PES curvature information from frequency calculations.

It is often best to search for transition states with a lower level of theory and lower basis set to get a good approximation to what the TS should look like.


Frequency Calculations

d 2EdRAdRB

= d 2 ˆ H dRAdRB

= ψ d 2 ˆ H dRAdRB

ψ

The Helmann Feynman theorem allows us to calculate forces and frequencies. The second derivative matrix is known as the Hessian:

The eigenvalues of this matrix are proportional to the frequencies squared (using the Harmonic approximation, and dividing by the reduced mass of the vibrational mode).

ν = km

E

Normal modes are the eigenvectors of the Hessian and describe the relative displacement vectorson each atom for the different vibrational modes.

Gaussian calculates vibrational frequencies for a given molecule at a particular structure, level of theory, etc.These calculations scale one order greater than energy calculations.


Frequency Calculations

Frequencies often are systematically high for a given level of theory and can thus be systematically corrected.This is usually done with a multiplicative scaling factor. For HF the scaling factor is about 0.89.

The best estimate for the vibrational frequencies using a HF frequency calculation is 0.89 times the HF frequencies.

Each level of theory will have its own scaling factor. The scaling factor can be determined by calculating thefrequencies for a molecule where the frequencies have been accurately measured and assigned.

Often, certain modes, such as inversion and dihedral torsion will not scale and these modes are oftennot corrected.


Frequencies

Frequencies can be used for several different things:1) They can be used to identify molecular species2) Help make assignments3) Determine the zero point energy and make ZPE corrections to energies4) Determine vibrational partition functions to determine vibrational entropy and pre exponential factors.5) Characterizing stationary points.6) Obtaining force constants to aid in TS searches.

E

ZPE


Levels of Theory

Gaussian can perform many different types of calculations. We’ll not go into all of these, but we will describe several of them during the course. We list them know so that you will be aware of them to usethem as you wish:

HF (UHF, RHF)MP2MP4CISCISDCISD(T)CASSCFQCISDQCISD(T)CCSDCCSD(T)G2

BLYPB3LYPB3PW91


Many Body Perturbation Theory

Configuration Interaction (CI) methods allowed us to systematically correct the Hartree-Fock wave functionby mixing in higher order excitations of the HF reference determinant.

The CI method is:Variational: Its energy is an upper bound to the ground stateNOT size consistent: The energy of a system does not scale with size. For example, two separated molecules will be different if calculated separately and then added, versus calculated together.

Is not: 2X

The reason CI is not size consistent is because truncating the expansion at some level of excitation from the HFreference wave function will exclude some configurations from the composite system that will be included in the fragments.

For example, double excitations out of both molecules simultaneously will be a quadruple for the composite CI,and not included in CISD. It is included as 2 times the double excitations in the fragment system.


Many Body Perturbation Theory

MBPT methods are an alternative way to systematically correct the Hartree-Fock wave functionby mixing in higher order excitations of the HF reference determinant.

MBPT is:NOT Variational: Its energy is not an upper bound to the ground state energy.Size consistent: The energy of a system scales with size.

Moller-Plesset many body perturbation theory is a way of correcting the HF ground state for its lack ofcorrelation energy (just as CI was). However, for MP MBPT methods, excited state character to includedynamic correlation is introduced using perturbation theory, rather than mixing in excited states variationally.

E

Order of perturbation


MP MBPT

Perturbation theory starts with a zero order wave function and energy and then improves it using a perturbation. In the case of Moller Plesset theory, the zero order Hamiltonian is taken as the sum of Fock Operators:

HHF

= fii=1

n∑

Where each Fock operator is the one electron kinetic energy, interaction with the nuclei, and interactionwith the average field of its neighbors:

fi = −∇i

2

2− ZA

riAA∑ + ν HF(i)

The HF ground-state wavefunction is an eigenfunction of HHF with eignvalue E(0) given by the sum of theoccupied single electron orbital energies. The perturbation is the difference between the full electronicHamiltonian and the mean field approximation:

H (1) = H − fii=1

n∑


MP MBPT

The first order correction to the energy is the expectation value of H(1) for the HF wavefunction, which isalready included in the HF energy:

E 1( ) = ψ 0 H(1)ψ 0

The leading correction to the HF ground state energy is then the second order perturbation energy:

E 2( ) =ψ J H (1) ψ 0 ψ 0 H (1) ψ J

E 0( ) − EJJ ≠0∑

Here, J refers to HF excited determinants.

Thus, perturbation theory mixes in excited state character depending on how the perturbation couplesthe excited state, and the difference in energy between the excited state and the ground state.

Since no truncation is done, MP theory is size consistent. Since the corrections are not mixed in variationally,the method is not variational.

Difference in energy

Coupling between states

You might notice that Brillouin’s theorem has something to say about which determinants have non-zeromatrix elements and contribute to the second order correction (the doubly excited configurations).


Many-body Perturbation Theory

Notes:

Perturbation theory has also been used to develop expressions for higher order corrections to the ground state wavefunction. These expressions can be derived in a similar way to the waythe lower order corrections are developed (see a general text on Quantum Mechanics) or using graph theory (see Szabo and Ostlund).

Second order perturbation theory: MP2 scales as N5. Improves bond energies, and frequenciesand transition state energies significantly.

MP4 further improves on MP2, although it is highly computationally intensive and generally onlypractical for small systems.


Unrestricted Wavefunctions

Often times Gaussian (and Quantum chemistry in general) allows one to make a choice as to whether to use a restricted or unrestricted wavefunction.

A restricted wavefunction doubly occupies each spatial orbital with two electrons.

A restricted open-shell wavefunction is one which doubly occupies the closed shell orbitals of anopen shell system.

An unrestricted method allows the electrons are not constrained to the same spatial orbitals. For example,UHF H2 would allow each electron to have its own spatial wavefunction, whereas the RHF method wouldforce the two electrons into the same bonding orbital.

Unrestricted methods have lower energies than restricted methods (since there are now more variationalcoefficients).

Unrestricted methods are not eigenfunctions of S2, the total spin operator and are often called spin contaminated(how far from a pure spin state the unrestricted method is). This is generally not a big problem.

Unrestricted methods should be used for open shell systems.

Some unrestricted methods:UHF, UMP2, UB3LYP, etc.

Density Functional Theory Lecture 11 Page 1

Density Functional Theory

Until now we have discussed methods which start with the Hartree-Fock approximation to obtain spinorbitals, which are then used to construct configuration state functions (using either CI or MBPT).

In principle CI and many body perturbation theory are exact when expanded to infinite order, however, thesemethods are both computationally intensive, even when only expanded to the first few terms (CISD, or MP2,for example).

Density Functional Theory is an alternative to HF based wave function methods which have become very popular due to there price-performance (they are quite accurate considering their computational efficiency).

Configuration State Functions:A Slater determinant, or linearcombination of Slater determinants.

HF

Singles Triples

DoublesCI or MBPT usedto mix in higher orderCSFs


DFT

Density Functional Theory is based on the concept that energy of an electronic system is a function ofthe electron density.

The Hohenberg-KohnTheorem (1965) proved that the ground state energy, and all other ground state electronic properties were uniquely functions of the electron density.

The basis idea is that the electronic wave function uniquely determines all electronic properties,including the potential energy operator, and the electron density. Consequently, if there is a one-to-onemapping between the wave function, the potential function, and the electron density, we can writethe energy as a functional of the electron density.

ρ r( ) ↔ υ r( ) ↔ Ψ0 E ρ( )↔ E Ψ0( )

Problem: the HK theorem doesn’t tell us the functional form of the dependence of energy onthe electron density.


DFT

The energy is written as the sum of the 1) kinetic energy, 2) the electron-nuclear interactions, 3) theelectron-electron interactions and the 4) exchange-correlation energy (unknown).

Ekin = − h2

2mψ 1

* r1( )∫i

∑ ∇12ψ 1 r1( )dr1

Ee−n = − ZIe2

riI∫

i, I∑ ρ1 r1( )dr1

Ee−n = 12

e2

r12ρ1 r1( )ρ2 r2( )dr1dr2∫

EXC = EXC ρ( )

E = Ekin + Ee−n + Ee−e + EXC


DFT

To determine the energy we use the variational principle to minimize the energy with respect tovariational parameters (basis set coefficients typically):

− h2

2m∇1

2 − ZIe2

riI+

ρ r2( )e2

r12dr2 + VXC (r1)∫

I∑

ψ i (r1) = εiψ i(r1)

The density is determined using:

ρ r( ) = ψ i (r)2

i∑

The reason DFT is computationally efficient is because integrals over two electrons to determinethe coulomb and exchange energy are now replaced by a one electron integral for coulomb, anda functional VXC for exchange.

Also note that DFT explicitly takes into account correlation in the exchange-correlation function.This allows solution similar to the Fock mean field approach althoughthe question is how to determine VXC, and how accurate is it.

VXC is determined using:

VXC =δEXC ρ[ ]

δρ


DFT

Note that just like HF, the wave function (and density) depends on the Hamiltonian, and theHamiltonian depends on the wave function (as does the density). Thus, like HF DFT methodsgenerally employ a Self-Consistent Field procedure.


Choose Coefficients

(Starting Guess)

Calculatedensity

CalculateXC

potential


New coefficientsAnd new

XC

Convergence

yes

no


Concepts determiningbasis set choice are thesame as for wave functionmethods.


Local Density Approximation

One method that has been used for determining the exchange correlation function is to use the LDA. Here, theexchange energy is approximated by the exchange energy per electron of a uniform electron gas with the same density as the local density at point r in the molecule.

This gives an analytic form for the exchange energy and potential since the exchange energy of a uniform electron gas can be solved exactly. The problem is that this is not a great approximation for for the exchangeenergy of a molecule.


BLYP, PW91, etc.

Other methods which improve upon LDA have been developed which use more sophisticateddescriptions of the exchange correlation functional. BLYP and PW91 are such methods.

They are fast, and quite accurate. Energies of stable states and frequencies are accurate. TransitionState energies are underestimated.

The Becke3LYP (B3LYP) method is similar to BLYP in that it also uses the LYP functional, butit mixes in Hartree-Fock exchange in an amount that is determined empirically by fitting to the G2 molecular set. This method is known as a hybrid since it combines HF and DFT. Since it usesHF it scales with the 4th power of the size of the system.

B3LYP improves upon HF, PW91, and BLYP with better energies, especially transition state energies(remember DFT underestimates EA whereas HF overestimates EA).

The approximate ordering of methods in terms of errors:

EHF > EPW91 > EBLYP > EMP2 > EB3LYP > ECISD > EMP4 > EQCISD > ECCSD

The approximate ordering of methods in terms of CPU time:

TCCSD > TQCISD > TMP4 > TCISD > TMP2 > TB3LYP > THF > TPW91 > TBLYP


DFT Notes

DFT methods generally:Underestimate activation barriers (2-5 kcal/mol typically for B3LYP).Overestimate frequencies by 1 to 5 percent.Predict good geometries, but using more optimization steps.Are very good methods for d-block transition state metals.

New DFT methods are being developed which have the accuracy of the best methods,yet are faster than B3LYP.

Practical Points Lecture 12 Page 1

Relative Energies

Relative Energies between stable molecules:• Moderately sensitive to basis set• Moderately sensitive to level of theory

Propylene Rotational Barriers:

∆E(0° → 180°) HF BLYP B3LYP MP23-21G* 1.77 1.90 1.89 1.726-31G(d) 2.07 2.06 2.08 1.966-311G(d) 2.04 1.90 1.94 1.846-31G(d,p) 2.07 2.04 2.06 1.98


Bond Energies

Bond Energies:• Moderately sensitive to Basis Set• Very sensitive to level of theory• HF underestimates bond energies, as do most HF based ab initio methods• DFT methods often overestimate bond energies

C-H Bond Energy in CH4 H-H Bond Energy

E(CH3-H) HF BLYP B3LYP MP23-21G* 86.64 113.64 114.24 101.086-31G(d) 86.56 111.89 112.84 103.906-311G(d) 85.16 110.34 111.27 103.396-31G(d,p) 87.23 112.60 113.50 108.99

E(H-H) HF BLYP B3LYP MP23-21G* 81.93 110.21 110.43 92.786-31G(d) 81.80 109.41 109.77 92.676-311G(d) 80.58 107.63 108.13 91.786-31G(d,p) 84.63 111.08 111.69 101.15


Transition State Energies

Transition State Energies:• Activation barriers are generally underestimated by DFT methods• Overestimated by HF and HF based ab initio methods• Barrier is reduced by increasing the basis set• Barriers are very sensitive to level of theory• Possibly the most difficult property to calculate accurately.

Silane Dehydrogenation Activation Barrier:

Barrier HF BLYP B3LYP MP23-21G* 80.48 57.74 62.05 68.256-31G(d) 80.00 56.88 61.33 67.576-311G(d) 79.26 56.91 61.24 66.656-31G(d,p) 76.68 54.71 48.92 65.39


Vibrational Frequencies

Vibrational Frequencies:• Moderately affected by basis set• Moderately affected by level of theory

Si-H Sym Stretch modes HF BLYP B3LYP MP23-21G* 2405.762 2211.433 2273.865 2351.27166-31G(d) 2396.526 2183.956 2250.447 2319.76716-311G(d) 2361.178 2172.692 2235.81 2293.36536-31G(d,p) 2368.431 2176.732 2239.663 2343.3492


Geometries

Molecular structure is not usually very sensitive to basis sets. However, in strained systems, and transition states basis sets may affect the structure moderately.

Molecular structure is usually not very sensitive to level of theory. HF, for example gives quite accurate geometries for most systems. Again, transition state structures may be more sensitive to level of theory.

Note: In various cases the level of theory won’t be appropriate to the molecule: For example, HF will not describe transition metals well. In those cases structure will be very sensitive to method.


Split Basis Sets

Basis Sets can often be Split to maximize the accuracy of the method on the chemically activepart of the molecule.

• Add additional functions to atoms that have bonds being stretched, or broken• Add additional functions to atoms that are donating or accepting charge

Note: Adding additional functions to DFT may (usually) make activation barriers worse.

Note: Splitting the basis set and Split Valence basis sets are not the same thing.


Constrained Optimizations

In many cases, geometry optimizations should be done with constraints.

This often includes cases where the cluster represents a portion of a larger molecule where the rest of the molecule added rigidity to the structure.

Constraints can also be used to • narrow a transition state search and • scan potential energy surfaces.

Systematic Corrections Lecture 13 Page 1

Systematic Corrections

Many of the errors inherent in the approximations needed to make ab initio methods practicalare systematic.

Various correction methods have been developed to estimate the size of the errors due to these approximations, and to determine approximate corrections for the various methods.

Corrections have been developed for:• vibrational frequencies• basis set truncation• correlation energy

Each of these corrections can be applied to a specific calculation individually, or combined toform a method, for example the G2 method.


Vibrational Frequencies

Scaling factors can be derived to correct vibrational frequencies. This is done by performingcalculations on small molecules with well characterized frequencies similar to the molecule of interest to obtain a multiplicative scaling factor:

S = 1J

ν j (exp)

ν j (theory)j∑

The best estimate for the experimental frequency of the unknown frequency is then:

ν j (scaled) = S ×ν j (theory)

Zero point energy corrections can also be based on scaled frequencies. Note that relative energies will not be very sensitive to scaling because both states will be upward corrected by the ZPE correction.


Corrections for Basis Set Truncation

All calculations must be performed at a particular basis set.

• The larger the basis set, the more accurate the calculation (usually). However the calculation also becomes more expensive with bigger basis sets.

• Various studies have noted that the largest errors in calculating total energies usually come from the finite size of the basis set.

• We also know that successively increasing the size of the basis set leads to smaller and smaller improvements in energy. Thus, the asymptotic convergence of the total energy with basis set size can be used to determine an extrapolated estimate for the energy for the complete basis set.

Various schemes have been developed to do this and all involve a series of calculations with different basis sets (and sometimes at higher level methods) to determine the extrapolation.


Correlation Energy Corrections

Not only can correction schemes be developed to determine the effects of basis set truncation, butextrapolation to higher order methods that include more correlation energy can also be performed.

Extrapolation schemes consist of a series of calculations at higher levels of theory to estimate theconvergence point of the total energy with level of theory. A series might include HF, MP2, MP4and QCISD.

HF is used as the base method.MP2 is used to include second order corrections to the energy.MP4 is used to approximate higher order corrections.QCISD is used to determine how well MP4 approximated the higher order corrections and thus howto include the higher order corrections.

The best calculations would involve performing high level methods (say QCISD or CCSD) at very large basis sets. Because it is impractical to do this calculation we approximate the best calculationby correcting for both basis set and correlation energy together. Several methods have been developed to do this.

CBS Methods: CBS-4 and CBS-Q

G1, G2, and G3


CBS Methods

CBS: Complete Basis Set extrapolation methods.

CBS-4

1) HF/3-21G(d) opt for geometry.2) HF/3-21G(d) freq for ZPE.3) HF 6-311+G(3d2f,2p) For base energy.4) MP2/6-31+G with basis set extrapolation to correct basis set and correlation to 2nd order.5) MP4 /6-31+G(d,p) to estimate higher order corrections.

The CBS-Q method is similar:

CBS-Q

1) MP2/6-31G(d) opt for geometry.2) HF/6-31G(d) freq for ZPE.3) HF 6-311+G(3d2f,2p) For base energy.4) MP2/6-31+G with basis set extrapolation to correct basis set and correlation to 2nd order.

Basis set extrapolation done with more points that CBS-45) MP4 /6-31+G(d,p) and QCISD(T) to estimate higher order corrections.

Higher order corrections for spin contamination and core correlation often added.


The G2 Method

The G2 method is similar to the CBS-Q and CBS-4 methods and also involves a series of calculations:

1) HF/6-31G(d) opt for geometry.2) HF/ 6-31G(d) freq for ZPE.3) MP2/6-311G(d,p) opt from HF opt in 1. Use this geometry for all later jobs.4) MP4/6-311G (d,p) Single point for base level energy.5) MP4/6-311+G (d,p) Single point to correct base for diffuse functions6) MP4/6-311G (2df,p) Single point to correct base for polarization functions (set to zero if greater than from

step 5)7) QCISD(T)/6-311G(d,p) Correction for correlation energy8) MP2/6-311+G(3df,2p) Correct for non-additive corrections in 5, and 69) High level corrections and Spin correction

QCISD and CCSD Methods Lecture 14 Page 1

The QCISD and CCSD Methods

Configuration Interaction methods truncate the CI expansion and a certain excitation, for exampleCISD truncates the CI expansion to include the Hartree Fock ground state determinant and singles anddoubles excitations from the HF reference configuration.

∑∑ Ψ+Ψ+Ψ=Ψrsab

rsab

ra

ra TTc

,0

,000

The finite truncation leads to the problem of size consistency where the CI energy does not scale with the size of the system.

E 2He( ) ≠ 2E(He)

For example, the energy of two isolated He atoms should be the same as two times the energy of each He atom calculated separately. This is not true for CISD.

The finite truncation leaves out certain configurations which are physically simultaneous excitations ofeach fragment which are formally higher order excitations.


Size Consistency Example: He

∑ ∑++=ar ar ba

barsabTrs

abcba

baraTr

acba

bac ϕϕϕϕ

ϕϕϕϕ

ϕϕϕϕ

ψ ˆˆ0

HF Singles Doubles+ +

∑ ∑++=ar ar

rsabTrs

abcraTr

ac

dcba

dcba

dcba

dcba

c

dcba

dcba

dcba

dcba

dcba

dcba

dcba

dcba

ϕϕϕϕϕϕϕϕϕϕϕϕϕϕϕϕ



ψ ˆˆ0

HF Singles Doubles+ +

Notice that only half (2 of 4) of the occupiedorbitals are replaced at the doubles level.


Size Consistency Example: He

As the number of He atoms grows, the percentage of orbitals excited at the singles and doubleslevels decreases. Thus, the CISD energy approaches the HF energy as the system becomes infinitelylarge.

HF Single Double

This is formally a double excitation, althoughmost of the He is in the HF ground state.

This is also formally a double, although most of the He is in the HF ground state.


QCISD

QCISD is a method which similar to CISD, but which adds excited state determinants whichare formally triples, and quadruples but which arise due to simultaneous singles and doublesor simultaneous doubles excitations.

The terms that are added are:

Disconnected Triples:

T1T2

This term accounts for triples that arise due to simultaneous singles and doubles excitations.

Disconnected Quadruples:1

2T2( )2

This term accounts for quadruples that arise due to simultaneous doubles excitations. Thefactor of one half is included to remove the double counting of pairs of doubles.

These are the terms which lead to size consistency.


QCISD

Notes:

• The QCISD method is extremely computationally intensive. This should not be usedon systems with over 6-8 heavy atoms on a fast workstation.

• To achieve the best accuracy, QCISD should be basis set extrapolated.

• The QCISD(T) method involves adding direct or connected triples to QCISD.This leads to a large improvement in energies, especially for transition states and excitedstates.

• QCISD is intermediate to the couple-cluster CCSD method in that itincludes many of the same terms included in CCSD, but not all the CCSDterms.

• QCISD approaches the accuracy of CCSD, but is somewhat less computationally intensive.


Coupled Cluster Theory

The coupled cluster method is one of the most accurate methods available today. It also involvesadding terms left out in the CI truncation to make the method size consistent. However, it does thisusing the wave function:

Ψ = exp ˆ T ( )Ψ0

ˆ T = ˆ T 1 + ˆ T 2 + ˆ T 3 + ...

The operator T is just the sum of all the excitation operators:

The T operators produce excited configuration determinants from lower order determinants and includeunknowns for the relative amount of the configuration to be included in the overall wave function (theseare the variational parameters equivalent to the CI expansion coefficients.

The CCSD method uses the wave function:

Ψ = exp ˆ T 1 + ˆ T 2( )Ψ0 = 1+ ˆ T 1 +1

2ˆ T 1

2...

1+ ˆ T 2 +

1

2ˆ T 2

2...

Ψ0

Note that this form includes all disconnected triples and quadruples through the terms;

ˆ T 1 ˆ T 21

2ˆ T 2 ˆ T 2

1

3ˆ T 1 ˆ T 1 ˆ T 1

1

2ˆ T 1 ˆ T 1 ˆ T 2


CCSD

As in QCISD(T), we can also include triples in approximately using the CCSD(T) method.

Triples could also be included exactly using:

Ψ = exp ˆ T 1 + ˆ T 2 + ˆ T 3( )Ψ0

This is called CCSDT.

Coupled Cluster theory has become the method of choice when very high accuracy is required.

Beware that this method is extremely expensive, but is appropriate when very small systemsare to be studied at high accuracy.

Geometries and frequencies are not usually computed at the CCSD level due to the computational expense:Typically the CCSD energies are computed at geometries obtained with lower level methods.

a brief introduction to gaussian

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