basics&of&computaonal& chemistry&

Basics of Computa.onal Chemistry

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Theory, Computation and Modeling

Theory

Computa.on

Modeling

Rules postulated to govern the behavoir of physical systems.

HΨ=EΨ E=mc2

Can be experimentally tested to valide the range of applicability. Not universally aplicable does not mean we should dispose the theory.

The deliberate introduc.on of approxima.ons into a more general theory. OFen incorporates constants from experimental data.

Essen.als of Computa.onal Chemistry, Chris Cramer , 2004 (Wiley)

Use of digital technologies to solve the mathema.cal equa.ons behind a theory or a model.

Theory or Computation?

Abhik Ghosh, 2003 (hSp://www.chem.uit.no/KJEMI/0)

What can be computed?

Structure

In principle if mesurable, predictable.

The rela.ve posi.on of the atoms that form the molecule. The minimal energy structure (which preserves connec.vity) is our target. In computa.on oFen we compute a molecule isolated from its medium.



Poten.al Energy Surface

In principle if mesurable, predictable.

All possible arrangements of a set of atoms against its poten.al energy.


(We assume Born-‐Oppenheimer approxima.on)

Heat of forma.on and combus.on. Complexa.on energies Hydrogen bond strength


Chemical Proper.es


-‐  Single-‐molecule proper.es:

•  Spectra (IR, NMR, EPR, UV, microwave, etc.) •  Ioniza.on poten.al (IP) and Electron Affini.es (EA) •  Total energies (useless) •  Thermodynamic data (H, G).

-‐  Composed proper.es:

•  Thermodynamic data (ΔH, ΔG) -‐> reac.vity. •  Equilibrium constants (from !) and rate constants.

Kine.c Isotop Effect pKa values -‐  Non-‐observable proper.es:

•  Aroma.city or bond orders.

Cost and Efficiency

The value of Computa.onal Chemistry


-‐  Green chemistry.

-‐  Cost of reagents vs. cost of technology.

-‐  It will never replace experiment.

-‐  It can provide informa.on not available experimentally, e.g., mechanis.cs.

-‐  Computa.on with respect to experiment can be:

1.  Post facto: Decides over an experimental ambiguity. 2.  A priori: Guides the experiment. 3.  Avoid Experiment: Goes where experiment cannot go (e.g. Dangerous experiment or dangerous chemistry)

Cost and Efficiency

The cost of Computa.onal Chemistry


-‐  Computa.onal chemistry can be expensive (.me and money-‐wise).

-‐  Some calcula.ons request CPU effort, other big amounts of MEMORY. Some also require LARGE DISK storage.

-‐  The job of the computa.onal chemist is knowing what CAN be calculated and HOW it can calculated, pugng emphasis on REDUCING the cost.

-‐  Nowadays it is not only maSer of choosing the right method but also the right soFware. Many choices available:

-‐  Bo=omline: Computa.onal chemist should know: the theory (predicted cost), the methods (to choose the op.mal) and the soFware (the best implementa.on).

hSp://en.wikipedia.org/wiki/Category:Computa.onal_chemistry_soFware

First calculations

Input: Minimal components

There are five minimal components of an input:

-‐  The method: So far we will restrict to B3LYP.

-‐  The basis set: Together with the basis it determines the accuracy of our calcula.on.

-‐  The structure: The ini.al structure that we must calculate

-‐  The charge: The total charge of our molecule

-‐  The mul.plicity: 2S+1, being S the number of unpaired electrons.

Input: Op.onal components

By default the program perform a single-‐point calcula.on of the energy. We may also request op.miza.on, loca.on of TS, calcula.on of proper.es, spectra, etc.

La gran mayoria de los cálculos de estructura electronica se basan en la aproximacion orbital y en la expresion de los orbitales moleculares como combinacion lineal de orbitales atomicos OM-CLOA.

Uno de los factores que mas influye en la calidad de los resultados de un calculo es el conjunto de orbitales atomicos que se introduzca.

Conjuntos de Base

El conjunto de base o Basis set es el conjunto de funciones monoelectrónicas en las que se expanden los orbitales moleculares. Como generalmente son funciones centradas en los núcleos, podemos entenderlas como los orbitales atómicos de los átomos que componen la molécula. Éstos orbitales atómicos son los que se combinaran para formar los orbitales moleculares.

Hay dos tipos principales:

Slater-type orbital:(STO):

Gaussian-type orbital:(GTO):

Si m+n+l = 0 la función es de tipo S Si m+n+l = 1, la función es P Si m+n+l = 2 , la función es D , etc De esta manera surgen 6 funciones D, 10 funciones F, etc. Podremos controlar que Gaussian use funciones d y f puras o cartesianas escribiendo 5d 7f o 6d 10f, respectivamente

Conjuntos de Base

Las funciones tipo STO describen mejor la densidad electrónica cerca de los núcleos, pero son computacionalmente ineficientes. Los GTO son computacionalmente mas manejables pero es necesario introducir combinaciones lineales de ellas (primitivas) para describir los electrones internos (core).

El programa Gaussian puede usar únicamente funciones GTO.

GTO (verde) vs STO (rojo)

Base Mínima: STO-3G Incluye una función GTO (formada a partir de tres primitivas) para cada capa. Atomos: H-Xe

H C S Fe 1s 1s 2s2p 1s 2s2p 3s3p 1s 2s2p 3s3p 3d 4s4p

Split-valence Incluye dos (doble-zeta), tres (triple-zeta) o más funciones de base para describir los orbitales de valencia

Doble zeta: 6-31G. Atomos: H-Kr H C S Fe

1s 2s 1s 2s2p 3s3p 1s 2s2p 3s3p 4s4p 1s 2s2p 3s3p 3d 4s4p 4d 5s5p

Triple-zeta: 6-311G. Atomos: H-Kr H C S (*) Fe(*)

1s 2s 3s 1s 2s2p 3s3p 4s4p 1s 2s2p 3s3p 4s4p 5s5p 6s6p

1s 2s2p 3s3p 3d 4s4p 4d 5s5p 5d 6s6p 7s 8s 9s

Full double–zeta: D95. Átomos: H-Cl Doble set de funciones para todas las capas

H C S (*) 1s 2s 1s 2s2p 3s3p 4s 1s 2s2p 3s3p 4s4p 5s5p 6s

Conjuntos de Base Existen innumerables conjuntos de base para cada átomo. Algunos típicos que están incluidos en Gaussian son los siguientes

(*) Para los átomos del segundo y tercer periodo se aplican otras reglas no generales

Correlation consistent: cc-pVXZ donde X=D,T,Q,5,... Inclusión progresiva de funciones polarización (momento angular mayor que la capa de valencia).

Funciones de polarización Los conjuntos de bases 6-31G, 6-311G o D95, etc pueden ser ampliados con funciones de momento angular mayor que la de la capa de valencia (de tipo p para el H, de tipo d para átomos del segundo periodo). Por ejemplo

6-31G* = 6-31G(d) Añade funciones d a los átomos pesados (no H) 6-31G** = 6-31G(d,p) Añade funciones d a los átomos pesados y p al H

También se puede incluir polarización múltiple. Por ejemplo, la base 6-311G(2df,2pd) incluye, además de las funciones propias de la base 6-311G, 2 sets de funciones d y un f a los átomos pesados y 2 sets p y un d al H.

Conjuntos de Base

Átomos cc-pVDZ cc-pVTZ cc-pVQZ H 2s,1p 3s,2p,1d 4s,3p,2d,1f

B-Ne 3s,2p,1d 4s,3p,2d,1f 5s,4p,3d,2f,1g Al-Ar 4s,3p,1d 5s,4p,2d,1f 6s,5p,3d,2f,1g

Funciones difusas: Las bases 6-31G, 6-311G, D95 pueden ser ampliadas con funciones Gaussianas con exponentes pequeños que ayudan a describir la densidad electrónica a distancias lejanas del núcleo. Son especialmente indicadas para el calculo de aniones e interacciones moleculares. Por ejemplo:

D95+, 6-31+G Añade funciones difusas s y p a los átomos pesados D95++, 6-31++G Añade difusas s y p a los átomos pesados y s al H

Se suelen incluir conjuntamente con las funciones de polarización de manera que podemos usar combinaciones de bases tipo 6-31+G(d) , 6-311++G** o D95++**, etc...

La manera de incluir funciones difusas a los conjunto de base correlation consistent (cc-pVXZ, X= D, T, Q, 5) es mediante el prefijo aug- . Así, tendremos las bases: aug-cc-pVDZ, aug-cc-pVTZ, aug-cc-pVQZ, etc...

Conjuntos de Base

Gaussian ofrece la posibilidad al usuario de introducir las funciones de base que desee usar para el cálculo. De esta manera podemos usar bases externas, personalizadas o incluso podemos utilizar conjuntos de bases diferentes para diferentes átomos de la molécula. Para ello debemos introducir el keyword GEN en lugar del nombre de la base interna de Gaussian e introducir la base en el propio fichero de input de Gaussian, después de la especificación de la geometría (y dejando una línea en blanco). El formato con el que el keyword GEN funciona es exactamente el mismo que el que el keyword gfinput escribe en el output de Gaussian la información de la base. Existe un formulario web con un conjunto muy extenso de funciones de base (muchas de ellas no están internamente en Gaussian). http://www.emsl.pnl.gov/forms/basisform.html Los pasos a seguir son: •Elegir el conjunto de funciones de base deseado de la lista •Especificar para qué elementos se requiere (incluir símbolos atómicos separados por comas) •Seleccionar el programa de calculo para el que se requiere (en este caso hay que escoger la opción Gaussian94). •Desactivar la opcion Optimize General Contractions (no aplicable para Gaussian) •Submit!

El formulario genera la información de las funciones de base en el mismo formato que Gaussian la acepta a través del keyword GEN.

Conjuntos de Base

First calculations

The ini.al structure

There are different ways to determine the ini.al structure:

-‐  Obtain the structure from: •  Experiment (X-‐ray) •  Research paper (Suppor.ng Informa.on). •  Previous (more approximate) calcula.on. •  Database Comp. Chem. Comparison and Database Benchmark (small molecules):

hSp://cccbdb.nist.gov/

-‐  Construct the structure: •  With the help of a chemistry graphic interface (gaussview, chemcraF, etc.) •  Cartesian Coordinates. •  Z-‐matrix coordinates.

Sistemes de Coordenades

Coordenades Internes Expressen posicions relatives dels àtoms en funció de distàncies interatomiques, angles d’enllaç i angles díedres

Coordenades Cartesianes Expressen posicions absolutes dels àtoms en l’espai 3D respecte a un centre de coordenades arbitrari.

L’energia del sistema es i ha de ser translacional i rotacionalment invariant i també es independent del sistema de coordenades utilitzat

3N - 6 coordenades internes

(3N-5 per sistemes lineals)

3N coordenades Cartesianes

Per un sistema molecular format per N nuclis

Altres sistemes de coordenades Coordenades Redundants Utilitzades internament per alguns programes de càlcul (Gaussian) per raons d'eficiència computacional

O1 H2

H3 a1

r1 r2

O1

H2 1 r1

H3 1 r2 2 a1

Z-‐matrix

1er átomo

2o átomo: distancia

3er átomo: distancia + angulo

Ángulos de enlace

3N -‐ 6 coordenadas internas

Coordenades Internes

O1 O2

H3 H4

a1 a2 r1

r2 r3

O1

O2 1 r1

H3 1 r2 2 a1

H4 2 r3 1 a2 3 d

O1 O2

H3 H4

d

Angulo entre el plano que con.ene los átomos 4-‐2-‐1 y el plano que con.ene los átomos

2-‐1-‐3

1er átomo



4o átomo en adelante: distancia + ángulo +

diedro

Angulo diedro Distancias Ángulos de

enlace Átomos

3N -‐ 6 coordenadas internas

Z-‐matrix

C1 C2

H3

H5

F4

H6

a1 a2

a3 a4

r1 r2

r4

r3

r5

C1 C2 1 r1

H3 1 r2 2 a1

F4 2 r3 1 a2 3 0.0 H5 1 r4 2 a3 3 H6 2 r5 1 a4 3 180.0

Distancias Ángulos de enlace

Ángulos diedros

Átomos

1er átomo




diedro

Átomos en el mismo plano en disposición cis

Z-‐matrix (molécula plana)

Átomos en el mismo plano en disposición trans

Ángulo diedro impropio (átomos 2 y 3 no directamente

conectados)

180.0

C1

H2

H3

H5

H4

a1

a2

a3

r1

r2

r4

r3

C1 H2 1 r1

H3 1 r1 2 a1

H4 1 r1 3 a1 2 120.0 H5 1 r1 3 a1 2

1er átomo


3er átomo: distancia + ángulo


diedro

Z-‐matrix

Ángulo diedro puede ser nega.vo

-‐120.0

120.0

-‐120.0

Todas las distancias Iguales por simetría Se puede usar la misma variable

Ángulos de enlace iguales.

Angulo tetraédrico 109.471º

arccos(-‐1/3)

(molécula no plana)

C1 C2

X3 r1

r3

Z-‐matrix (átomos dummy)

H4 H6

X5

C1 C2 1 r1

X3 1 1.0 2 90.0

H4 1 r3 3 90.0 2 180.0 X5 2 1.0 1 90.0 3 H6 2 r3 5 90.0 1 180.0

Ángulos de enlace Su valor debe estar entre 0º y 180º

La posición de los átomos dummy no necesita el uso de

variables. Se debe incluir la

distancia y angulos de forma explicita en la z-‐

matrix 0.0

r3

Química Computacional

Mecànica Molecular Mecànica Quàn.ca

Mètodes semiempírics

Mètodes ab ini.o

• L’energia potencial del sistema ve donada per una funció analítica anomenada camp de forces. • No té en compte els electrons. Es considera cada àtom com una partícula. • A cada àtom se li assigna com a mínim un radi i una carrega neta parcial. • Els enllaços es tracten con si fossin molles amb constants de força i distància d’equilibri determinades. No es poden trencar enllaços. • Altres interaccions no enllaçants com electrostàtiques o de van der Waals s’inclouen típicament al camp de forces mitjançant la Llei de Coulomb i potencials tipus Lennard-Jones, respectivament.

Es basen en la resolució aproximada de l’equació de Schrödinger electrònica (ESE), ja que en la immensa majoria dels casos treballa sobre la aproximació de Born-Oppenheimer

• Per a la resolució aproximada de la equació de Schrodinger electronica s’utilizen paràmetres empírics obtinguts prèviament a partir de dades experimentals o càlculs acurats (ab initio). • El fet d’incloure paràmetres simplifica també el cost computacional del càlcul

S’aplica les lleis de la Física Clàssica per modelar els sistemes moleculars.

• Proporcionen la funcio d’ona electrònica i l’energia potencial del sistema de l’estat fonamental electronic i/o altres estats excitats.

• Resolució aproximada de l’ESE a partir dels postulats de la MQ. • No es necessiten dades experimentals. L’energia i funció d’ona del sistema depenen només de la posició dels nuclis. • El grau de sofisticació en la resolució de l’ESE determina el cost computacional del càlcul

Energy Calculation

Solve Schrödinger Equa.on

HΨ=EΨ

The energy is the central quan.ty in quantum mechanics. The Schröndinger Equa.on gives means to calculate it:

Nature is thriFy in all its ac.ons Maupertuis (1744)

where

H=T+Vee+VNe+VNN

an Ψ is a well-‐behaved func.on:

-‐ Single-‐valued. -‐ Con.nuous. -‐ Finite. -‐ Its square is integrable. -‐ An.symmetric with interchange of all coordinates.

Energy Calculation

The role of E

E is a basis set and method dependent quan.ty. Thus, the absolute energy is a oFen a useless quan.ty. (Chemistry works with rela.ve energies)

However, E is oFen used to op.mize the Ψ (Ψ"Emin), in varia.onal methods.

We will use Gaussian during this course. Bookmark this page:

hSp://www.gaussian.com/g_tech/g_ur/g09help.htm

Exercise 1: Compare energy values of H2 at HF/6-‐31G*, HF/6-‐311G*, B3LYP/6-‐31G* and CISD/6-‐31G*

Energy Calculation

Self-‐Consistent Field (SCF)

Electronic energy

Convergence aSained Total angular momentum of spin: <S2>

Assumes that Ψ is a Slater determinant. The SCF is needed in prac.ce for most methods: HF, MP2, CISD, CCSD and KS-‐DFT.

It changes if open-‐shell or DFT methods used.

Energy Calculation


Set the convergence criteria. Usage:

-‐ Conver=N (10-‐N) SCF(conver=8) [default (N=6)] -‐ Tight (N=8) SCF(.ght) [default in geometry op.miza.on]

Use in Gaussian

Guess & checkpoint file

-‐ We can start the calcula.on informa.on %chk=name.chk [specified in the first line] -‐ We can read the Ψ from previous calc. guess=read [useful to speed up the process]

Energy Calculation


First of all: Usage

-‐ Check the geometry makes sense. -‐ Change the ini.al Ψ guess: guess=core guess=indo guess=huckel

SCF op.ons:

-‐ Try quadra.c convergence SCF(XQC) or SCF(QC) [uses second deriva.ves: expensive!] -‐ Increase the number of cycles SCF(MaxCycle=256) [default MaxCyc=128, only if exceeded] -‐ Use Damping SCF(damp,ndmap=10) [Uses damping factor 5-‐15] -‐ Use Level ShiF.ng SCF(VshiF=1) [increases HOMO-‐LUMO gap by X a.u., stabilizes first cycles useful for some DFT problema.c cases]

Convergence problems

Potential Energy Surface (PES)

It arises from Born-‐Oppenheimer approxima.on. It plots the energy (all contribu.ons but nuclei kine.c energy) with respect to nuclei coordinates (could be any representa.on: Cartesian, internal, normal modes, etc.)

There are a few relevant cri.cal points (zero gradient) of the

PES:

Minima characterize the stable geometries.

Saddle points of first order (one nega.ve Hessian eigenvalue)

correspond to transi.on states (TS).

•  Aproximación de la función por serie de Taylor hasta segundo orden

•  Condición de gradiente cero

•  Paso de Newton-Raphson (exacto para funciones cuadráticas)

•  Metodo Iterativo Newton-Raphson

Newton-Raphson

•  Cálculo inicial de la matriz de segundas derivadas (Hessiana) –  Por defecto estimación por métodos empíricos –  Opt = CalcFC cálculo exacto –  Opt = CalcHFFC cálculo exacto a nivel Hartree-Fock –  Opt = Readfc lectura de la hessiana de un cálculo previo (fichero chk)

•  Aproximación de la matriz Hessiana en cada punto –  Metodos de actualización de la hessiana (aproximada) a partir de la posición y

gradientes de interaciones anteriores. Por defecto, método BFGS –  Después de varias iteraciones la Hessiana actualizada puede diferir

notablemente de la exacta!

Pseudo-Newton-Raphson

•  Método Newton-Raphson - Opt=CalcAll Cálculo de la matrix hessiana en cada punto

Geometry Optimization OPT

•  Elección del sistema de coordenadas –  Por defecto, coordenadas redundantes –  Opt=z-matrix Usar variables de la Z-matrix de input –  Opt= cartesian Cartesianas –  Opt=Modredundant Añadir o modificar variables redundantes. –  [Permite optimización con restricciones, scan, etc..]

•  Criterios de Convergencia (default threshold RMS Force 3·10-4 au) •  Opt=Loose Criterio de convergencia muy laxo. (10-3 au) •  Opt=Tight, Opt=VeryTight Criterio más estricto. (10-5 & 10-6 au) [Only DFT: Use Int=Ultrafine (more accurate energy calculation) also]

Maximum structural change in an atom in the last two itera.ons

first deriva.ve

Geometry Optimization

•  Máximo número de iteraciones excedido –  Volver a optimizar desde punto de gradiente o energía menor. –  Incrementar número de iteraciones Opt=Maxcyc=N [El número por defecto depende del número de variables del sistema]

•  Convergencia lenta u oscilante –  Disminuir el tamaño del paso (stepsize) Opt=Maxstep=N [Por defecto N=30, disminuir a 1-5]

•  Cambio de grupo puntual durante optimización –  Deshabilitar la simetría Opt=Nosymm

Cuando el proceso de optimización no converge…

•  Check molecular structure / charge & multiplicity. •  Check symmetry group (symmetrize with GaussView/ChemCraft) •  Avoid too long/too short distance.

TIPS

Geometry Optimizaton

Convergence aSained with 4 YES Gaussian output

> < <

<


Is the structure a minimum?

-‐ We cannot guarantee its an absolute minimum of the PES. (unless we have the whole PES)

-‐  The first deriva.ves should be zero -‐> sta.onary point (opt).

-‐  The second deriva.ves should be nega.ve.

-‐  The calcula.on of second deriva.ves is done by reques.ng a frequency analysis (freq).

-‐ All second deriva.ves nega.ve only guarantees that it is a LOCAL minimum.


EXERCISE 2:

Op.mize the structure of the following isomers of and compare their rela.vite energies at the HF/6-‐31G* and B3LYP/6-‐31G* levels and calculate its IR spectrum.

1,2-diaminoethen-1-ol 2,2-diaminoethen-1-ol

Geometry Optimizaton EXERCISE 2:

1,2-diaminoethen-1-ol 2,2-diaminoethen-1-ol

Electronic energy Conformer HF B3LYP ΔE(Kcal/mol) ΔE(Kcal/mol) trans 1,2 -262.94150 -264.5047552 4.34 3.15 cis 1,2 -262.94842 -264.5097804 0.00 0.00

2,2 -262.93952 -264.5020935 5.58 4.82

Free energy Conformer HF B3LYP ΔG(Kcal/mol) ΔG(Kcal/mol) trans 1,2 -262.871442 -264.442356 3.55 2.42 cis 1,2 -262.877103 -264.446217 0.00 0.00

2,2 -262.869796 -264.440223 4.59 3.76

Transition State (TS)

Point in the PES with zero gradient and only one nega.ve Hessian eigenvalue.

The nega.ve eigenvalue corresponds to the direc.on that leads to reactants and products, i.e. the reac.on coordinate.

The path that connects the reactants and the products through the TS is known as the reac.on path.

TS Optimization (I)

To locate a TS in Gaussian we should first of all have some knowledge of the chemical reac.on. Start by op.mizing the reactants and the products, consider if we have some addi.onal knowledge about the TS (Hammond postulate) and design a good guess of the TS structure using R o P geometries. Then we will perform a gaussian calcula.on using the keyword opt=(calcfc,TS,noeigentest).

Foresman and Frisch, Exploring Chemistry with Electronic Structure Methods, (Gaussian, Inc., PiSsburgh, PA, 1996).

TS loca.on is more an art than a science. We should use our knowledge and chemical intui.on. However, a good guess is more than the work. In difficult cases we can use the keywords:

Readfc: Read frequencies (useful if you have frequencies from lower level calc). Calcfc: Performs freq calcula.on at the first point. Calcall: Performs freq calcula.on at every opt point (only for drama.c cases).

TS Optimization

EXERCISE 3:

Study the SN2 reac.on of Cl-‐ with CH3Cl at the HF/3-‐21+G*. Op.mize the reactants, products and TS. Perform frequency analysis of the TS.

SN2 reac.on 20 min (altogether)

ΔG♯= 2 Kcal/mol AH♯= 0 Kcal/mol

TS Optimization (II)

The localiza.on of a TS is confirmed by the presence of one and only nega.ve imaginary frequency.

Some.mes is not easy to design a good guess for the TS structure.

Gaussian provides two keywords to automa.cally generate a star.ng molecular structure for a TS op.miza.on based upon reactants and products structures. It is called STQN method and is implemented under the keywords QST2 (needs R+P) and QST3 (needs R+P+ guess TS).

Foresman and Frisch, Exploring Chemistry with Electronic Structure Methods, (Gaussian, Inc., PiSsburgh, PA, 1996).

These vector-‐following methods are par.cularly useful for symmetric reac.ons or reac.ons where the structure ‘around the ac.ve part’ does not change or changes symmetrically along the reac.on path.

TS Optimization

EXERCISE 4:

Study the reac.on of HF with CH3 to produce CH4 + F using QST2 and QST3 at the HF/3-‐21+G* (count the number of electrons!) Perform frequency analysis of the TS.

20 min (altogether)

guess TS PROD REAC

QST2

QST3

Op.mize them separately and merge the structures

Do not op.mize them simultaneously! (idem for PROD)

TS Characterization

Once we have obtained a converged TS search (zero gradient):

1.  Calcula.on of second deriva.ves (Hessian). Freq calcula.on.

2.  Check the number of nega.ve eigenvalues (imaginary frequencies)

•  One: We got a TS (but maybe not the one we want!) •  More than one: Saddle point of order n.

•  Zero: We got a minimum structure (unlikely if searching for TS)

3.   Check our transi.on vector (i.e. animate the normal mode associated to the TS, the nega.ve eigenvalue). It should correspond to the mo.on associated to our reac.on (i.e. this mo.on leads to reactants and products).

4.  Not clear yet? Trace the reac.on path: -‐> IRC

If we have more than one nega.ve eigenvalue, animate them and choose those not associated to our reac.on. By following the corresponding vectors we can reach the “minimum” energy structure (i.e. turn the nega.ve eigenvalues into posi.ve ones).

Conformational structure

EXERCISE 5:

Calculate the energy and frequency of staggered and eclipsed conforma.on in ethane. Give the energy gap. (Choose yourself the method/basis)

12 min (altogether)

Scan

Amb la opció SCAN podem anar fent càlculs puntuals d’energia a mesura que anem canviant algun paràmetre geomètric. (si volguéssim op.mitzar cada un dels punts podríem afegir la opció opt).

-‐ Avantatges – Em permet anar seguit una coordenada, que pot ser de gran importància en una reacció. Ens donarà una idea del que fa el sistema.

-‐ Desavantatges – -‐ S’ha d’u.litzar la molècula definida a par.r de una Z-‐matrix (definida de forma adequada) -‐ Compte amb el canvi de simetria -‐ NOSYMM

Scan

C

C 1 r1

Cl 1 r2 2 a1 H 1 r3 2 a2 3 d1 H 1 r3 2 a2 3 d2 Cl 2 r2 1 a1 3 d3 H 2 r3 1 a2 6 d4 H 2 r3 1 a2 6 d5

Definim variables: I la que volem scanejar li afegim: D3 0. 18 10. 18 steps (10º each)

Scan

Càlcul Puntual d’energia en cada punt

# Metode/Base Scan Nosymm

Op.mització en cada pas

# Metode/Base OPT Nosymm

Reaction Path Intrinsic Reac.on Coordinate (IRC)

•  Minimum reac.on path energy (thermodata excluded) that connects reactants and products. Express in mass-‐weighted Cartesian coordinates ( x m1/2 )

•  It needs the op.mized structure of the Transi.on State and the Hessian

•  To obtain each structure along the reac.on path towards products/reactants the program performs a minimiza.on restricted to a sphere of radius n/2 where n is the stepsize.

Stepsize = R/2

IRC Gaussian Options

CalcFC: Calculates the Hessian at the first point.

RCFC: Reads the Hessian from chk file (usually from previous TS characteriza.on).

ReadIso: Read the isotops for each atoms (default: most common). MaxPoint=n: Walk n points along the IRC in each direc.on (6 both by default).

Forward: Go towards reactants.

Reverse: Go towards products.

StepSize=n: Stepsize in the IRC (n x 0.01au).

NoSymm: Ignore Symmetry (needed when TS has different symmetry than R o P).

Analysis of a Chemical Reaction

1.  Op.mize R and P (opt).

2.  Characterize R and P (freq & visual inspec.on). 3.  Op.mize TS. (opt=TS,noeigentest,calcfc or QST2/QST3).

4.  Characterize the TS (freq & mode anima.on).

5.  Calculate the ΔH, ΔG, ΔG± (from thermodata).

6.  Calculate the IRC and check connec.on: R-‐-‐TS-‐-‐P.

Electromagnetic spectrum

Born-Oppenheimer Approximation

Born-Oppenheimer Approximation

V(D) poten.al energy surface

Potential Energy Surface (PES)

It arises from Born-‐Oppenheimer approxima.on. It plots the energy (all contribu.ons but nuclei kine.c energy) with respect to nuclei coordinates (could be any representa.on: Cartesian, internal, normal modes, etc.)

There are a few relevant cri.cal points (zero gradient) of the

PES:

Minima characterize the stable geometries.

Saddle points of first order (one nega.ve Hessian eigenvalue)

correspond to transi.on states (TS).

Vibrational Analysis

Harmonic approximation

•  It gives rise to vibra.onal modes. •  There are 3N-‐6 normal modes (3N-‐5 for linear molecules) •  Each normal mode is concerted mo.on of many atoms. •  Each normal mode is harmonic oscillator.

•  The energy is quan.zed (νP) •  The energy gap is constant (1/2 ωP) •  There is zero point energy (ZPE)

ωP is the harmonic frequency

Normal Modes of -CH2-

Normal Modes: Computation (ADVANCED MATERIAL)

No.ce that nega.ve Hessian eigenvalues lead to imaginary frequencies. Therefore, the TS is characterized by one imaginary frequency.

Normal Modes with Gaussian (ADVANCED MATERIAL)

We can read these force constants and follow the procedure described in the previous slide to “manually” obtain the frequencies

(This is, however, out of the scope of this course)

Vibrational analysis with Gaussian

Aqer op.miza.on (otherwise it makes no sense!) we can perform a freq calcula.on to assess the nature of our sta.onary point. If it is minimum it will the eigenvalues of the Hessian will posi.ve and, therefore, all the frequencies (normal modes) will be posi.ve.

! ! ! ! ! !1 2 3!! ! ! ! ! !B B B!

Frequencies -- 175.8455 298.5285 503.0039! Red. masses -- 1.6404 2.3145 1.3847! Frc consts -- 0.0299 0.1215 0.2064! IR Inten -- 0.0695 0.0031 12.6208! Raman Activ -- 8.6959 11.0215 3.1110! Depolar (P) -- 0.7499 0.5825 0.7500! Depolar (U) -- 0.8571 0.7362 0.8571!

They are sorted by the value of the frequency, thus if the lowest (1) is posi.ve our molecular structure is a minimum of the PES.

Gaussian Ouput:

Vibrational Analysis

Frequencies sorted from lowest to largest:

We can depict a IR spectra by represen.ng the IR frequencies on the X axis, against the IR intensity in the Y axis. Many program will even reproduce the shape of the spectrum.

Keyword: freq Gaussian Input:

Normal Modes with Gaussian

AFer op.miza.on (otherwise it makes no sense!) we can perform a freq calcula.on to assess the nature of our sta.onary point. If it is minimum it will the eigenvalues of the Hessian will posi.ve and, therefore, all the frequencies (normal modes) will be posi.ve.

! ! ! ! ! !1 2 3!! ! ! ! ! !B B B!

Frequencies -- 175.8455 298.5285 503.0039! Red. masses -- 1.6404 2.3145 1.3847! Frc consts -- 0.0299 0.1215 0.2064! IR Inten -- 0.0695 0.0031 12.6208! Raman Activ -- 8.6959 11.0215 3.1110! Depolar (P) -- 0.7499 0.5825 0.7500! Depolar (U) -- 0.8571 0.7362 0.8571!

They are sorted by the value of the frequency, thus if the lowest (1) is posi.ve our molecular structure is a minimum of the PES.

Other sta.onary points: A transi.on state (TS) will have one nega.ve eigenvalue (i.e. un imaginary frequency); all other cases are saddle points in the energy surfaces and, in principle, their are not so relevant to chemical reac.vity.

En una geometría de equilibrio la diagonalización de la matriz de constantes de fuerza da lugar a 6 valores propios igual a cero (5 para una molécula lineal). Los vectores propios asociados corresponden a los movimientos de traslación y rotación. Puesto que en el proceso de obtención de la estructura de equilibrio el gradiente no es exactamente cero (y debido a que la separación entre movimientos rotacionales y vibracionales no es total), se observan desviaciones respecto al cero en estos valores propios. Son lo que Gaussian etiqueta como low vibrational frequencies

Las transiciones entre niveles vibracionales se dan en la zona del infrarojo del espectro y las frecuencias vibracionales hármonicas son esenciales para interpretar el espectro. La intensidad de las senales en un espectro de absorcion IR vienen dadas por la magnitud del momento dipolar de transicion (que da lugar a las reglas de selección).

Fuerza del oscilador

Analisis vibracional

Thermochemistry

Water molecule (Output from Gaussian)

€

U 0 =Uelec0 +Uvib

0 ≡ ε0 +εZPE

€

U =U 0 +Utrans +Urot +Uvib ≡ ε0 + Etot

€

H =U + RT ≡ ε0 +Hcorr

€

G = H −TS ≡ ε0 +Gcorr

AFer a frequency analysis of the sta.onary points (TS or minima) we can retrieve important thermochemistry data.

We have rota.onal, transla.onal and vibra.onal correc.ons to the energy.

Entalphy and Free Energy

Thermodynamical data for the following reac.on (1 atm and 298.15K):

Data obtained using the data in Thermochemistry aDer a freq analysis.

Free Energy & Enthalpy

Reac.on enthalpy is given as the enthalphy energy of products minus reactants.

(Gibbs) Free Energy is given as the free energy of products minus reactants.

G

El Método Hartree-Fock (I) La principal estrategia para la resolución de la ecuación de Schrödinger electrónica es el método de Hartree-Fock (HF) o del Self-Consistent Field (SCF). Las claves son: - Aplicación de la Aproximación Orbital: el movimiento de cada electrón del sistema viene descrito por una función que depende de las 3 coordenadas del espacio: orbital molecular. - Cuando se especifica el spin del electrón estamos hablando de spinorbital molecular. - Los orbitales moleculares a su vez se expresan como una combinación lineal de un conjunto de funciones de base típicamente centradas en los núcleos que hacen el papel de orbitales atómicos. Aproximación MO-LCAO. El objetivo es el de determinar los coeficientes óptimos de dicha combinación lineal para cada orbital molecular. Esta aproximación no es parte del método Hartree-Fock pero es ampliamente utilizada en la actualidad por la mayoría de programas de cálculo. - La función de onda aproximada del sistema multielectrónico es un Determinante de Slater, formado por los spinorbitales moleculares que ocupan los electrones del sistema. El determinante de Slater cumple por construcción el Principio de Antisimetria de Pauli y es por tanto una buena función de onda para el sistema.

El método Hartree-Fock se basa en considerar la función de onda polielectrónica como un Determinante de Slater y determinar los coeficientes de los orbitales moleculares ocupados aplicando el Principio variacional.

-El punto de partida es expresar la energía del sistema como valor esperado de la función de onda de prueba. -La aplicación del método variacional implica imponer que la variación de la energía respecto a los parámetros variacionales (coeficientes de los orbitales moleculares) sea cero. Se impone también la restricción de que los orbitales moleculares que forman el Determinante de Slater sean ortogonales entre si

El resultado es una ecuación de valores y funciones propias de un operador monoelectrónico: el operador de Fock. Las funciones propias del operador de Fock son los orbitales moleculares

Operador de Coulomb

El Método Hartree-Fock (II)

La particularidad de este operador es que su parte bielectrónica depende de la forma de los orbitales ocupados, que a su vez son sus propias funciones propias. Esto implica que la resolución de las ecuaciones de Hartree-Fock se lleva a cabo mediante un proceso iterativo

Operador de Intercambio

El hecho de expresar los orbitales moleculares como combinación lineal de los orbitales atómicos (funciones de base) transforma las ecuaciones de valores y funciones propias de Hartree-Fock en su equivalente algebraico:

El Método Hartree-Fock (III)

Matriz de densidad

La matriz F es la representación del operador de Fock monoelectronico en la base de orbitales atómicos (funciones de base). Es por tanto una matriz simétrica de dimensión M (número de funciones de base)

La matriz S es la matriz de solapamiento de las funciones de base. En los cálculos para moléculas esta base no es ortogonal por lo que la matriz S difiere de la identidad.

La solución a la ecuación matricial anterior son las matrices C y E. La matriz C es una matriz de dimensión M que contiene en columnas los coeficientes de la combinación lineal que determinan los orbitales moleculares.

El Método Hartree-Fock (IV)

Restricted (R): The spa.al part of the spinorbitals is the same for electrons α and β. It applies to systems with an even number of electrons and closed-‐shell structure. The wave func.on is eigenfunc.on of S2 operator, so the spin is well defined. It may also be used to open-‐shell systems, with the RO (Restricted Open-‐Shell) formalism, but in this case the orbital energies are not uniquely defined.

Unrestricted (U): The spa.al part of the spinorbitals can be different for α and β electrons. It is usually applied to open-‐shell systems, but can be always used. The wave func.on is eigenfunc.on of the Sz operator (which only determines the difference in the number of α and β electrons), but it is not in general an eigenfunc.on of the S2 operator. Thus, the total spin may not be well defined (only the average value) and thus the wave func.on may contain spin contamina.on (the WF is an effecFve average of configura.ons of different total spin)

The Slater determinants can be built from either restricted or unrestricted spinorbitals.

It is important to note that the expression of the spinorbitals as a product of the spa.al and angular part is only exact if the Hamiltonian does not contain spin-‐dependent terms. Thus, it will not be exact if spin-‐orbit coupling would be included

Restricted vs Unrestricted

O2 Singlete: RHF/STO-3G Spin total, S = 0 Multiplicidad de Spin, 2S+1 = 1 Valor esperado de S2, S(S+1) = 0

Energía SCF = -147.550985858

Ener

gia

σ1s

πx

σ*1s

σ2s σ*2s

πy σ2pz

πx*

El ultimo orbital ocupado (HOMO) en realidad debería ser degenerado (πx*, πy*). Sin embargo esto no pasa en este cálculo Puesto que hemos forzado que los electrones estén apareados el programa elige en cual de los orbitales π coloca el par de electrones (en este caso el πx*). La consecuencia es que se rompe la degeneración de los otros orbitales π y ahora el πx tiene una energía menor que el πy

O2 Triplete: ROHF/STO-3G Spin total, S = 1 Multiplicidad de Spin, 2S+1 = 3 Valor esperado de S2, S(S+1) = 2

Energía SCF = -147.632087505

Ener

gia

σ1s

πx

σ*1s

σ2s σ*2s

πy σ2pz

πx* πy*

En este caso, hemos forzado que dos electrones estén desapareados (eligiendo la multiplicidad de spin = 3) Ahora no se rompe la degeneración de los orbitales π y el diagrama de orbitales moleculares tiene una estructura mas familiar

O2 Triplete: UHF/STO-3G

Valor esperado de S2, S(S+1) = 2.003 Energía SCF = -147.63387085

Ener

gia

σ1s

πx

σ*1s

σ2s σ*2s

πy σ2pz πx* πy*

Orbitales α

Orbitales β

σ1s

πx

σ*1s

σ2s σ*2s

πy σ2pz

α β

El cálculo UHF hace que los orbitales que describen los electrones α y β puedan ser diferentes (y en este caso lo son). Esta libertad extra hace que baje la energía total SCF pero a cambio el spin de la molecula no esta bien definido.La contaminación de spin en este caso no es muy importante (<S2> deberia ser exactamente 2 para un triplete)

O2 Singlete: UHF/STO-3G

Valor esperado de S2, S(S+1) = 1.00 Energía SCF = -147.605053623

Ener

gia

σ1s

πxy

σ*1s

σ2s σ*2s

σ2pz

πxy*

Orbitales α

Orbitales β

σ1s σ*1s

σ2s σ*2s

α β

πxy’

πxy’ σ2pz

πxy’* πxy

πxy’*

Este cálculo muestra como actúa la contaminación de spin. La energia total es menor que en el caso RHF, sin embargo, mirando el valor de <S2> podemos ver que el programa no nos proporciona una función de onda singlete (donde <S2>=0) A primera vista, las energía de los orbitales α y β parecen las mismas. Sin embargo mirando las contribuciones de los orbitales atómicos vemos que algunos orbitales estan intercambiados! Por eso <S2>≠0

Correlation Energy

Ecorr=EHF-‐EFCI

FCI

MP2, CASSCF, CCSD CISD, QCISD, etc.

Variational Principle

It uses the calculus of varia.ons for finding a func.on (Ψ) that minimize the value of the quan..es (E) on this func.on, oFen imposing some boundary condi.ons (well-‐behaving func.on). It is oFen used by many methods of quantum mechanics to find the “best” wavefunc.on and its associated energy.

In general this is a good method to find the wavefunc.on, despite nothing guarantees that the wavefunc.on that minimizes the energy is the best one to calculate proper.es other than the energy.

Hartree-‐Fock, configura.on interac.on (CI) or complete ac.ve-‐space (CAS) methods use this principle.

We say that a method is varia.onal if the energy obtained with this method is an upper bound to the full-‐CI (FCI) energy, which represents the exact energy.

€

Eexact ≤ Ψ0 Η Ψ0 = E0

€

Ψ0 ⇔ E0

Electron Correlation Hartree-Fock: Provides M molecular orbitals (MO) [2M spinorbitals (SO)] expressed as combinación of atomic orbitals (AO). These MO [SO] have an energy and the N/2 [N] with the lowest energy are occupied and the other M-N/2 [2M-N] are empty (virtual orbitals). The HF wavefunction consists of only one Slater determinant constructed from these occupied orbitals.

Post-Hartree-Fock methods: The wavefunctions is constructed as a linear combination of Slater determinants, constructed using the occupied and virtual HF orbitals.

Hartree-Fock Single Excitation Double excitation Nth excitation

...

Configuration interaction (CI)

The wavefunctions is constructed as a linear combination of Slater determinants exciting an increasing number of electrons (first one, two, etc.). It uses the variational method to optimize the coefficients {c0, etc.} multiplying the Slater determinants.

The full expansion (Full-CI or FCI) provides the exact energy but it too costly. In practice, people truncate the expression after a given number of excitations: CIS, CISD, CISDT, etc. The most popular CI method is CISD (CI singles and doubles) constructed by performing only mono- and bi-excitations. Nowadays is an obsolete method and very rarely used.

Post-HF methods (I)

€

Ndet =MN /2⎛

⎝ ⎜

⎞

⎠ ⎟

CI is a variational method. CI is not size-consistent except for FCI method. The cost of CISD scales like M6. It introduces dynamic correlation.

Size consistency If the energy of a system composed of two or more non-‐interac.ng fragments is equal to the energy of each fragment using a given method, that method is size consistent.

€

E(A…B;R→∞) ≠E(A) + E(B)

The property of size consistency is important to obtain a correct dissocia.on curve.

He dimer calculated using CID with a minimal basis set illustrates this Problem:

Computational Cost

It is important to know the computa.onal cost of a single-‐point calcula.on in terms of the size of our basis set (M), namely we want to know its scaling as Mn. (more sophis.cated counts include scaling in terms of occupied and virtual orbitals).

Hartee-‐Fock: M4

It implies that doubling the size, increases the computa.onal cost 16 (24) .mes.

CISD: M6

It implies that doubling the size, increases the computa.onal cost 64 (26) .mes.

Basis Z=1-‐2 Z=3-‐10

STO-‐3G 1 5

3-‐21G 2 9

6-‐31G* 2 15

6-‐311+G** 7 23

Number of func.on per atom

Perturbational Theory Møller-Plesset (MPn)

Based on classical perturba.on theory that studies the effects of small disturbances. It provides an approximate solu.on by exploi.ng a good exis.ng approxima.on (Y0,E0) and performing small correc.ons over it.

If the Hartree-‐Fock is used as zero-‐order approxima.on the theory is known as Møller-‐Plesset (MPn), where n indicates the order (i.e. the number of correc.ons). However, the correc.ons on the energy and the wavefunc.ons does not match each other, the correc.on on the wavefunc.on goes ‘ahead’ the energy correc.on:

E(0) = Σiεi Ψ(0) = ΨHF

E(0) +E(1) = EHF Ψ(0)+Ψ(1)

E(0) +E(1)+E(2) Ψ(0)+Ψ(1)+Ψ(2)

MP0

MP1

MP2

MP is not varia.onal and size consistent. MP2 scales like M5. MP3 like N6, etc.

Be aware: The perturba.onal series is not necessarily convergent!!!!

Coupled Cluster (CC) Theory

It uses an alterna.ve strategy to build a list of configura.on interac.ons to aSain ΨFCI:

where T is the cluster opera.on, defined as:

T = T1 + T2 + T3 + … + TN

where N is the number of electrons and T2 is an operator which generates biexcited determinants. The difference with CI steems from the fact that the final expansion is different, and it contains more excita.ons, enough to make CC size consistent. On the other hand it is not varia.onal. It is, by far, the most popular post-‐HF method. CCSD scales like M6.

CCSD(T)

It is called Gold Standard and is oFen used for benchmarking. It performs CCSD an a triple correc.on on top. It is not varia.onal either but it keeps size consistency. It scales like M7.

€

Ψ = eΤΨHF

Density Functional Theory (DFT)

The Ψ is essen.ally uninterpretable and too complicated to handle (4N coordinates). DFT suggests to use a much simpler ansatz: the density. The challenge is expressing the Hamiltonian as a func.on of the density (energy func.onal).

In short: we do not approximate the Ψ anymore, we use the ‘exact density’ and approximate the ‘Hamiltonian expression’.

Hohenberg y Kohn (1967) theorems: The ground-‐state density determines the external poten.al, thus any observable (including the E) can be determined by the density. The exact density minimizes the energy func.onal, i.e., it leads to the minimal energy.


The exact expression of the energy func.onal in terms of the density E=E[ρ] is not known and, therefore, there is a collec.on of proposal energy func.onals to be used. There is more than 70 func.onals suggested. In prac.ce no more than 20 are used in rou.ne calcula.ons: B3LYP, PBE, BP86, BLYP, TPSS, M06, M06X, OLYP, LDA, OPBE.

B3LYP (gas phase) and PBE (solid state) cover >90% of current DFT calcula.ons.


B3LYP (gas phase) and PBE (solid state) cover >90% of current DFT calcula.ons.

•  Are DFT func.onals varia.onal? No, they are not. •  Are DFT func.onals size consistent? Most of them, yes. •  Which is the scaling of DFT?

In general M3 – M4 (hybrids) but there are some excep.ons.

Why DFT so popular?

It includes (somehow) electron correla.on at very low price.

There is no other method with such small average (energy!) error at such low cost.

H2 energy

EXERCISE 6:

Compare energy values of H2 at HF/6-‐31G*, HF/6-‐311G*, B3LYP/6-‐31G* and CISD/6-‐31G*

We take R=0.7419 Å cc-‐pVQZ basis set.

HF -‐1.1334478 MP2 -‐1.1665654 CISD -‐1.1737965 MP3 -‐1.1716956 CCSD -‐1.1737965 MP4 -‐1.1731775 B3LYP -‐1.1805363 FCI -‐1.1737965

HF/6-‐31G*=-‐1.1267257 HF/6-‐311g*=-‐1.1279637

As we increase the basis set the energy improves

CISD/6-‐31G*=-‐1.1516858 CISD=FCI (exact method) B3LYP/6-‐31G*=-‐1.1754819 B3LYP is not varia.onal

(-‐1.1755 < 1.1517)

Now let’s try more methods

(units Hartrees)

He2 energy

EXERCISE 7: Compare energy values of He2 at 8Å with two separate He atoms

We take cc-‐pVQZ basis set.

He atom -‐2.8615142 -‐2.8969922 -‐2.9014374 -‐2.9022302 -‐2.9024109 -‐2.9024109 -‐2.9149814

He2 R=8Å -‐5.7230285 -‐5.7939845 -‐5.8028749 -‐5.8044604 -‐5.8048218 -‐5.8041768 -‐5.8299623

Size Consistency 0 0 0 0 0 0.4 Kcal/mol 0

HF MP2 MP3 MP4 CCSD CISD B3LYP

Exact Non-‐varia.onal Not size consistent

Convergent MP series

(units Hartrees)

Population Analysis

Wavefunction population analysis: Mulliken Population Natural Population Analysis (NPA) Löwdin Population Mayer bond orders

Electrostatic potential population analysis MK GHELPG

Electron density population analysis: AIM – Quantum theory of Atoms in Molecules

There is not a unique way to define an atom in a molecule. Therefore, the concept of atomic boundaries, atomic charges or, in general, atomic property is not well defined; it is an observable.

However, the old concepts of atomic charges and bond orders are deeply rooted in the chemistry community and often called ‘Population Analysis’. Among them, let us mention a few:

pop=(full)

pop=(nbo)

IOp(6/80)=1

Mulliken Population Analysis

•  It defines an atom as the set of atomic orbitals centered in that atom.

It takes a basis set and split its func.ons among the atoms.

•  Unlike other methods, it is solved analy.cally.

•  Suffers with func.ons of “non-‐atomic” character, e.g. polariza.on and, especially, diffuse func.ons.

Mulliken Population Analysis

where P is the density matrix and S the overlap matrix.

Mulliken Population Analysis (Gross Populations)

Electrons assigned to each atomic orbital

(AO) by using the fomula:

STO-‐3G 6-‐31++G(d,p)

Mulliken Population Analysis Mulliken (Atomic) charge

Electrons assigned to A

6-‐31++G(d,p)

STO-‐3G

QA = NA-‐ZA

Charge assigned to A

€

NA = ρµµ =1

M

∑

Löwdin Population Analysis Mayer Bond Order

V(H)=0.95

0.97 V(O)=δ(O,H2)+δ(O,H3)=0.949+0.949=1.898

V(H2)=δ(O,H2)+δ(H2,H3)=0.949+0.016=0.965 0.97 0.02

Valences

Methods in Comp. Chemistry

Method Variational? Size consistent? Scaling (M^n) HF Y Y 4 DFT N Usually 3-4 MP2 N Y 5 CISD Y N 5 CCSD N Y 6 MP3 N Y 6 MP4 N Y 7

M is the basis set size.

basics&of&computaonal& chemistry&

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