nneeenne V

M

1A

M

AB AB

BA

V

N

1i

N

ij ij

V

N

1i

M

1A iA

A

T

M

1A

2A

A

T

N

1i

2i R

ZZr1

rZ

2M1

21

H

Born-Oppenheimer Approximation TN=0, VNN=ct

Hartree-Fock equations

N

i

N

i

N

ij ij

M

A iA

AN

iielec rr

ZH

1 111

2 1

2

1

)(xΦ...)(xΦ)(xΦ

)(xΦ...)(xΦ)(xΦ

)(xΦ...)(xΦ)(xΦ

)(N!)r,...,r,(rΨΨΨ

NNN2N1

2N2221

1N1211

1/2N21

SDHF0

ijji

N

1i

N

1jijij

N

1ii

SDSD )K(J21

HΨHΨE

(x)dxΦv(x)21

(x)ΦH i2*

ii

212*j2j

121

*i1iij dx)dx(x)Φ(xΦ

r1

)(x)Φ(xΦJ

- Core integral (one-electron integral

- Coulombian integral (two-electron integral)

212*j2i

121j1

*iij dx)dx(x)Φ(xΦ

r1

)(x)Φ(xΦK

- Exchange integral (two-electron integral)

Looking for the spin-orbitals which give the best EHF under the orthonormalization conditions and using the method of the Lagrange multipliers one obtains (see Jensen pg.62-63)

Nif iiii ,1ˆ Hartree-Fock equations

Lagrangean multipliers are eigenvalues of the Fock operator:

(i)v1

-f HF

M

1A

2ii ˆ

2

1ˆ iAr

- one-electron operator

(i)vHFˆ

-Hartree-Fock potential

-The average repulsive potential experienced by the i-th electron due to the remaining N-1 electrons

-Replaces the complicated 1/r12 two-electron operator (electron-electron repulsion is taken into account only in an average way)

N

1j1j1j1HF )(xK-)(xJ)(xv ˆˆˆ

With:

212

22j1j dx

r1

)(xΦ)(xJ

- represents the average local potential at position x1 arising from an electron located in j

Operating with Jj(x1) we obtain:

)(xdxr1

)(xΦ)(x)(xJ 1i212

22j1i1j

depends only on the value of i(x1) Jj is a local operator

The second term of vHF(x1) is the exchange contribution to the Hartree-Fock potential:

)(xΦ)dx(xΦr1

)(xΦ)(x)Φ(xK 1j22i12

2*j1i1j

Kj(x1) exchanges the variables of the two spin-

orbitals

depends on the value of i on all points in space because it depends on the integrating variable Kj(x1) is a non-local operator

JijKij0 Jii=Kii no self interaction in Hartree-Fock approximation

Since fi depends on i HF equations must be solved iteratively (SCF procedure)

Initial guess Conv.?

vHF, f {i}

No

SD, EHF

Yes

N

1jijijiiiii )K(JHΦfΦε

Molecular orbital energies:

Total energy:

M

1A

M

AB AB

BAN

1i

N

1jijij

N

1iiT R

ZZ)K(J

21

E

For a closed-shell system in RHF formalism, the total energy and molecular orbital energies are given by (see Szabo and Ostlund, pag.83):

N/2

1i

N/2

1jijij

N/2

1ii )K(2JHE 2

N/2

1jijijii )K(2JH

Each occupied spin-orbital i contributes a term Hi to the energy  Each unique pair of electrons (irrespective of their spins) in spatial orbitals i and j contributes the term Jij to the energy Each unique pair of electrons with parallel spins in spatial orbitals ψi and ψj contributes the term –Kij to the energy

Examples:

a) b) c) d)

a) E=2H1+J12

b) E=H1+H2+J12-K12

c) E=H1+H2+J12

d) E=H1+2H2+H3+2J12+J22+J13+2J23-K12-K13-K23

1

2

3

Koopmans’ Theorem

j

ijijii )K(JHεii Φε

N

1i

N

1jijij

N

1ii

SDSD )K(J21

HΨHΨE

Total electronic energy:

Summing all εi:

iii j

ijijii )K(JHε i

iεE

why?because εi includes the Coulombian and exchange interactions with all the other electrons (also with εj). Similarly, εj will include also the interactions with εi, so that the interaction between the i-th and j-th electrons will be counted twice in the sum over orbital energies

What is the physical meaning of orbital energies?

So, what εi is?

Consider the ionization of the molecule (one electron removed from orbital number k) and suppose that no change of MO’s occurs during the ionization process

N

ki1i

N

1jijij

N

ki1i

i

N

1i

N

1jijij

N

1ii

k1NN )K(J

21

H)K(J21

HEE

k

N

1jkjkjk

k1NN ε)K(JHEE

nn

1-N

i

1-N

jijij

1-N

ii1-N V)K(J21

HE k

Koopmans’ theorem

vertical ionization potential (IP) photoelectron spectroscopy(obtained without optimizing the cation geometry)

adiabatic ionization potential is obtained when the geometry of the cation is optimized

physical meaning of εi: the i-th orbital ionization energy!

Hartree-Fock-Roothaan Equations

LCAO-MO

K

ii c1

i=1,2,...,K

{μ} – a set of known functions

The more complete set {μ}, the more accurate representation of the exact MO, the more exact the eigenfunctions of the

Fock operator The problem of calculating HF MO the problem of calculating the set cμi LCAO coefficients

)()()( 111 rcrcrf iii matrix equation for the cμi coefficients

Multiplying by μ*(r1) on the left and integrating we get:

111*

1111* )()()()()( drrrcdrrrfrc iii

1111* )()()( drrrfrF

- Fock matrix (KxK Hermitian matrix)

111* )()( drrrS

- overlap matrix (KxK Hermitian matrix)

KicScF iii ,...,2,1,

- Roothaan equations

FC=SC

KKKK

K

K

ccc

ccc

ccc

C

...

............

...

...

21

22221

11211

K

...00

............

0...0

0...0

2

1

More compactly:

where

-the matrix of the expansion coefficients (its columns describes the molecular orbitals)

The requirement that the molecular orbitals be orthonormal in the LCAO approximation demands that:

ijji Scc

The problem of finding the molecular orbitals {i} and orbital energies i involves solving the matrix equation FC=SC. For this, we need an explicit expression for the Fock matrix

Charge density

N/2

a

2a(r)Φ2ρ(r)

For a closed shell molecule, described by a single determinant wave-function

Nρ(r)dr

μν

*νμμν

μν

*νμ

(r)(r)P

(r)(r)

ρ(r)

2/*

2/**

2/*

2

)()(2)()(2

N

aaa

N

aaa

N

aaa

cc

rcrcrr

K

ii c1

The integral of this charge density is just the total number of electrons:

Inserting the molecular orbital expansion

into the expression for the charge density we get:

Where:

2/

*2N

aaaccP - elements of the density matrix

The integral of (r) is

NP(r)dr(r)P(r)(r)Pρ(r)drμν μν

μνμν*νμμν

μν

*νμμν Sdr

By means of the last equation, the electronic charge distribution may be decomposed into contributions associated with the various basis functions of the LCAO expansion.

μνμνP S -the electronic population of the atomic overlap distribution

-give an indication of contributions to chemical binding when and

centered on different atoms

Off-diagonal elements

Diagonal elements

- the net electronic charges residing in orbital μμμμSP

Population analysis ≡ allocate the electrons in the molecule in a fractional manner, among the various parts of the molecule (atoms, bonds, basis functions)                 -  Mulliken population analysis

Substituting the basis set expansion we get:

PS)PS()( trSPNdrr

Basis set functions are normalized Sμμ=1

Pμμ - number of electrons associated with a particular BF

- net population of φμ

 Qμ = 2PμSμ (μ≠)      overlap population

associated with two basis functions which may be on the same atom or on two different atoms

Total electronic charge in a molecule consists of two parts:

N N N

NQP

first term is associated with individual BF

second term is associated with pairs of BF

SPPq - gross population for φμ NqN

A

AnetA PZq

the net charge associated with the atom A

A B

AB Qq

total overlap population between atoms A and B

Mulliken population analysisFormaldehyde

#P RHF/STO-3G scf(conventional) Iop(3/33=6) Extralinks=L316 Noraff Symm=Noint Iop(3/33=1) pop(full)

Basis functions:

The summation is over occupied molecular orbitals

Example

)cc...ccc2(cP 18581252115151

μic

oc

1iνiμiμν cc2P

NSPρ(r)droc

μ

oc

νμνμν

μνμνSP

μνS

= sum over the line (or column) corresponding to the C(1s) basis function

= sum over the line (or column) corresponding to the O(2px) basis function

Atomic populations (AP)1 O 8.186789

2 C 5.926642

3 H 0.9432854 H 0.943285

Total atomic charges (Q=Z-

AP)1 O -0.186789

2 C 0.073358

3 H 0.0567154 H 0.056715

Molecular orbitals of formaldehyde (RHF/STO-3G)

Basis Sets

)(...)()(

)(...)()(

)(...)()(

)!(222

111

2/1

NKNjNi

Kji

Kji

xxx

xxx

xxx

N

)()()( jjiji rx

K

ii c1

with: {μ} – a set of known functions

for UHF wave-functions two sets of coefficients are needed:

orbitalsandforusedarefunctionsbasissamethe

c

c

K

ii

K

ii

1

1

if μ AO LCAO-MO

if μ AO LCBF-MO

Basis functions

• mathematical functions designed to give the maximum flexibility to the molecular orbitals• must have physical significance• their coefficients are obtained variationally

STO provide reasonable representations of atomic orbitalshowever they are not well suited to numerical (fast) calculations of especially two-electron integralstheir use in practical molecular orbital calculations has been limited

Slater Type Orbitals (STO)

- similar to atomic orbitals of the hydrogen atom- more convenient (from the numerical calculation point of view) than AO, especially when n-l≥2 (radial part is simply r2, r3, ... and not a polinom)

),(),,;,,,( 1 lm

rni YeNrrmln

STO – are labeled like hydrogen atomic orbitals and their normalized form is:

)exp( 1

2/131

1 rs

2exp

962

2/152

2

rrs

2exp

322

2/152

2

rx

xp

STOAdvantages:•Physically, the exponential dependence on distance from the nucleus is very close to the exact hydrogenic orbitals.•Ensures fairly rapid convergence with increasing number of functions.

Disadvantages:•Three and four center integrals cannot be performed analytically.•No radial nodes. These can be introduced by making linear combinations of STOs.

Practical Use:•Calculations of very high accuracy, atomic and diatomic systems.•Semi-empirical methods where 3- and 4-center integrals are neglected.

Gaussian Type Orbitals (GTO)-introduced by Boys (1950)-powers of x, y, z multiplied by-α is a constant (called exponent) that determines the size (radial extent) of the function

2re

nmlr zyxNezyxnmlg2

),,;,,,(

nmlrf zyxNezyxfnmlg22

),,;,,,,(

or:

N - normalization constantf - scaling factor

scale all exponents in the related gaussians in molecular calculations

l, m, n are not quantum numbersL=l+m+n - used analogously to the angular momentum quantum number for atoms to mark functions as s-type (L=0), p-type (L=1), d-type (L=2), etc (shells)

The absence of rn-1 pre-exponential factor restricts single gaussian primitives to approximate only 1s, 2p, 3d, 4f, ... orbitals.However, combinations of gaussians are able to approximate correct nodal properties of atomic orbitals

GTO – uncontracted gaussian function (gaussian primitive)

GTO - contracted gaussian function (gaussian contraction)

STO= GTO

GTOs are inferior to STOs in three ways:

GTO’s behavior near the nucleus is poorly represented.At the nucleus, the GTO has zero slope; the STO has a cusp. GTOs diminish too rapidly with distance. The ‘tail’ behavior is poorly represented.Extra d-, f-, g-, etc. functions (from Cart. rep.)may lead to linear dependence of the basis set. They are usually dropped when large basis sets are used.

Advantage:GTOs have analytical solutions. Use a linear combination of GTOs to overcome these deficiencies.

The first ten normalized gaussian primitives are:

)exp(2

),( 24/3

rrg s

)exp(128

),( 2

4/1

3

5

rxrg x

)exp(128

),( 2

4/1

3

5

ryrg y

)exp(128

),( 2

4/1

3

5

rzrg z

)exp(9

2048),( 22

4/1

3

7

rxrg xx

)exp(9

2048),( 22

4/1

3

7

ryrg yy

)exp(9

2048),( 22

4/1

3

7

rzrg zz

)exp(2048

),( 2

4/1

3

7

rxyrg xy

)exp(2048

),( 2

4/1

3

7

rxzrg xz

)exp(2048

),( 2

4/1

3

7

ryzrg yz

There are 6 possible d-type cartesian gaussians while there are only 5 linearly independent and orthogonal d orbitalsThe gs, gx, gy and gz primitives have the angular symmetries of the four corresponding AO.

The 6 d-type gaussian primitives may be combined to obtain a set of 5 d-type functions:

gxy dxy

gxz dxz

gyz dyz

22

2

4

3

22

1

yxyyxx

zyyxxzz

dgg

dggg

The 6-th linear combination gives an s-type function:

szzyyxxrr ggggg )(5 2/1

In a similar manner, the 10 f-type gaussian primitives may be combined to obtain a set of 7 f-type functions

GTOs are less satisfactory than STOs in describing the AOs close to the nucleus. The two type functions substantially differ for r=0 and also, for very large values of r.

cusp condition:for STO: [d/dr e-ξr]r ≠ 0

0]/[ 0

2

rredrd

for GTO:

With GTO the two-electron integrals are more easily evaluated. The reason is that the product of two gaussians, each on different centers, is another gaussian centered between the two centers:

),(),(),( 111 PGFsABB

GFsA

GFs pK RrRrRr

where:KAB=(2αβ/[(α+β)π])3/4exp(-αβ/(α+β)|RA-RB|2]The exponent of the new gaussian centered at Rp is: p=α+β

and the third center P is on line joining the centers A and B (see the Figure below)RP=(αRA+βRB)/(α+β)

The product of two 1s gaussian is a third 1s gaussian

allow a more rapidly and efficiently calculation of the two-electron integralsGTO

have different functional behavior with respect to known functional behavior of AOs.

contractions (CGF or CGTO)

L

pAp

GFppA

CGF d1

),()( RrRr L – the length of the contractiondpμ – contraction coefficients

How the gaussian primitives are derived?by fitting the CGF to an STO using a least square methodvarying the exponents in quantum calculations on atoms in order to minimize the energy

ExampleSTO-3G basis set for H2 molecule

Each BF is approximated by a STO, which in turn, is fitted to a CGF of 3 primitiveshydrogen 1s orbital in STO-3G basis setFor molecular calculations, first we need a BF to describe the H 1s atomic orbital

then: MO(H2) = LCBF

3 gaussian primitives:exponent coefficient0.222766 0.1543290.405771 0.5353280.109818 0.444636

23

22

21

3322113

1rrrGSTO

s ecNecNecN If we use a scaling factor:

223

222

221

3322113

1rfrfrfGSTO

s ec'Nec'Nec'N

βi=αif2

432

/i

i'N

712705402

43

./

! Using normalized primitives we do not need a normalization factor for the whole contractionIf the primitives are not normalized, we have to obtain a normalization factor. For this we use the condition:

131

31 GSTO

sGSTO

s |S

]deccNN

deccNNdeccNN

decNdecNdecN[F|

I

r)(

I

r)(

I

r)(

I

r

I

r

I

rGSTOs

GSTOs

6

232

5

231

4

221

3

23

2

22

1

21

3232

31312121

223

23

222

22

221

21

231

31

2

22

S=F2[I1+I2+I3+2I4+2I5+2I6]

2/3

1

21

212/3

1

2/321

213

1

21

21

0

2221

211 2

.)2(

.)2(4

1.44

21

cNcNcNdrrecNI r

But:

3/ 4 3/ 221 1

1 1

2 2N N

so that:211 cI

Analogously:222 cI 233 cI

2/321

2/3

2121321

2121

0

)(21214 )()(4

144

221

ccNNccNNdreccNNI r

4/32

4/31

2/34/3

2

4/3

121

222

NN

and thus:

212/321

4/32

4/31

2/3

4 )(

2ccI

312/331

4/33

4/31

2/3

5 )(

2ccI

322/332

4/33

4/32

2/3

6 )(

2ccI

Now,

322/332

4/33

4/322/5

312/331

4/33

4/312/5

212/321

4/32

4/312/52

322

21

2

)(2

)(2

)(2 cccccccccFS

Imposing that S=1 we obtain:

2/1

322/332

4/33

4/32

312/331

4/33

4/31

212/321

4/32

4/312/52

322

21 )()()(

2

1

ccccccccc

F

In the general case of a contraction of dimension n, the above expression become:

n

i

n

jjiji

ji

ccF1 1

4/34/3

2/3

2

The 1s hydrogen orbital in STO-3G basis set will be:

),()(3

11 rgcNr i

isi

with:

2

),(1r

iisieNrg

4/32

i

iN - normalization factors for primitives

3

1

3

1

4/34/3

2/3

2

i jjiji

ji

ccN

- normalization factor for the whole contraction (when un-normalized primitives or segmented contractions are used)

Summary

23

1

)(r

iii

i

eNcNr

αi βi ci Ni ci Ni

2.227660 3.425250 0.154329 1.794441 0.276934

0.405771 0.623913 0.535328 0.500326 0.267839

0.109818 0.168856 0.444635 0.187736 0.083474

N=1.0000002

Explicitly:

222 168856.0623913.0425250.31 083474.0267839.0276934.01)( rrrs eeer

If the exponents are not scaled:

222 109818.0405771.0227660.21 060453.0193973.0200560.01)( rrrs eeer

STO-3G basis set examplehttp://www.chem.utas.edu.au/staff/yatesb/honours/modules/mod5/c_sto3g.htmlThis is an example of the STO-3G basis set for methane in the format produced by the "gfinput" command in the Gaussian computer program. The first atom is carbon. The other four are hydrogens.  Standard basis: STO-3G (5D, 7F) Basis set in the form of general basis input:   1 0 S    3 1.00   .7161683735D+02   .1543289673D+00   .1304509632D+02   .5353281423D+00   .3530512160D+01   .4446345422D+00 SP   3 1.00   .2941249355D+01  -.9996722919D-01   .1559162750D+00   .6834830964D+00   .3995128261D+00   .6076837186D+00   .2222899159D+00   .7001154689D+00   .3919573931D+00 ****  2 0 S    3 1.00   .3425250914D+01   .1543289673D+00   .6239137298D+00   .5353281423D+00   .1688554040D+00   .4446345422D+00 ****  3 0 S    3 1.00   .3425250914D+01   .1543289673D+00   .6239137298D+00   .5353281423D+00   .1688554040D+00   .4446345422D+00 ****  4 0 S    3 1.00   .3425250914D+01   .1543289673D+00   .6239137298D+00   .5353281423D+00   .1688554040D+00   .4446345422D+00 ****  5 0 S    3 1.00   .3425250914D+01   .1543289673D+00   .6239137298D+00   .5353281423D+00   .1688554040D+00   .4446345422D+00 ****

Split valence basis setshttp://www.chem.utas.edu.au/staff/yatesb/honours/modules/mod5/split_bas.html

Valence orbitals are represented by more than one basis function, (each of which can in turn be composed of a fixed linear combination of primitive Gaussian functions). Depending on the number of basis functions used for the reprezentation of valence orbitals, the basis sets are called valence double, triple, or quadruple-zeta basis sets. Since the different orbitals of the split have different spatial extents, the combination allows the electron density to adjust its spatial extent appropriate to the particular molecular environment. Split is often made for valence orbitals only, which are chemically important.

3-21G basis setThe valence functions are split into one basis function with two GTOs, and one with only one GTO. (This is the "two one" part of the nomenclature.) The core consists of three primitive GTOs contracted into one basis function, as in the STO-3G basis set.

  1 0 //C atom S    3 1.00   .1722560000D+03   .6176690000D-01   .2591090000D+02   .3587940000D+00   .5533350000D+01   .7007130000D+00 SP   2 1.00   .3664980000D+01  -.3958970000D+00   .2364600000D+00   .7705450000D+00   .1215840000D+01   .8606190000D+00 SP   1 1.00   .1958570000D+00   .1000000000D+01   .1000000000D+01 ****  2 0 //H atom S    2 1.00   .5447178000D+01   .1562850000D+00   .8245472400D+00   .9046910000D+00 S    1 1.00   .1831915800D+00   .1000000000D+01 ****

6-311G basis set

Extended basis setsThe most important additions to basis sets are polarization functions and diffuse basis functions.

Polarization basis functionsThe influence of the neighboring nuclei will distort or polarize the electron density near a given nucleus. In order to take this effect into account, orbitals that have more flexible shapes in a molecule than the s, p, d, etc shapes in the free atoms are used.

An s orbital is polarized by using a p-type orbital

A p orbital is polarized by mixing in a d-type orbital

6-31G(d) a set of d orbitals is used as polarization functions on heavy atoms

6-31G(d,p) a set of d orbitals are used as polarization functions on heavy atoms and a set of porbitals are used as polarization functions on hydrogen atoms

Diffuse basis functionsFor excited states and anions where the electronic density is more spread out over the molecule, some basis functions which themselves are more spread out are needed (i.e. GTOs with small exponents). These additional basis functions are called diffuse functions. They are normally added as single GTOs. 6-31+G - adds a set of diffuse sp orbitals to the atoms in the first and second rows (Li - Cl). 6-31++G - adds a set of diffuse sp orbitals to the atoms in the first and second rows (Li- Cl) and a set of diffuse s functions to hydrogen. Diffuse functions can also be added along with polarization functions. This leads, for example, to the 6-31+G(d), 6-31++G(d), 6-31+G(d,p) and 6-31++G(d,p) basis sets.

 Standard basis: 6-31+G (6D, 7F) Basis set in the form of general basis input:   1 0 S    6 1.00   .3047524880D+04   .1834737130D-02   .4573695180D+03   .1403732280D-01   .1039486850D+03   .6884262220D-01   .2921015530D+02   .2321844430D+00   .9286662960D+01   .4679413480D+00   .3163926960D+01   .3623119850D+00 SP   3 1.00   .7868272350D+01  -.1193324200D+00   .6899906660D-01   .1881288540D+01  -.1608541520D+00   .3164239610D+00   .5442492580D+00   .1143456440D+01   .7443082910D+00 SP   1 1.00   .1687144782D+00   .1000000000D+01   .1000000000D+01 SP   1 1.00   .4380000000D-01   .1000000000D+01   .1000000000D+01 ****  2 0 S    3 1.00   .1873113696D+02   .3349460434D-01   .2825394365D+01   .2347269535D+00   .6401216923D+00   .8137573262D+00 S    1 1.00   .1612777588D+00   .1000000000D+01 ****

Number of primitives and basis functions for 1,2-Benzosemiquinone free radical with the STO-3G basis set

Primitives:atom C: nr.primitives = 15 x nr. atoms = 6 → 90atom H: nr.primitives = 3 x nr. atoms = 4 → 12atom O: nr.primitives = 15 x nr. atoms = 2 → 30

TOTAL: 132 GTO primitivesBasis functions:atom C: nr. BF = 5 x nr.atoms = 6 → 30atom H: nr. BF = 1 x nr.atoms = 4 → 4atom O: nr. BF = 5 x nr.atoms = 2 → 10

TOTAL: 44BF

Number of primitives and basis functions for 1,2-Benzosemiquinone free radical with the 6-31+G* (6-31+G(d)) basis set

Primitives:atom C: nr.primitives = 32 x nr. atoms = 6 → 192atom H: nr.primitives = 4 x nr. atoms = 4 → 16atom O: nr.primitives = 32 x nr. atoms = 2 → 64

TOTAL: 272 GTO primitivesBasis functions:atom C: nr. BF = 19 x nr.atoms = 6 → 114atom H: nr. BF = 2 x nr.atoms = 4 → 8atom O: nr. BF = 19 x nr.atoms = 2 → 38

TOTAL: 160BF

Effective core potentials (ECPs)•Core electrons, which are not chemically very important, require a large number of basis functions for an accurate description of their orbitals.•An effective core potential is a linear combination of specially designed Gaussian functions that model the core electrons, i.e., the core electrons are represented by a effective potential and one treats only the valence electrons explicitly.• ECP reduces the size of the basis set needed to represent the atom (but introduces additional approximations)•for heavy elements, pseudopotentials can also include of relativistic effects that otherwise would be costly to treat

p

1i

rξn0iECP

20iierd(r)UECP potentials are specified as parameters of the following equation:

where p is the dimension of the expansion di are the coefficients for the expansion terms, r0 is the

distance from nucleus and ξi represents the exponents for each term.

• Saving computational effort • Taking care of relativistic effects partly• Important for heavy atoms, e.g., transition metal

atoms

complexPd1.chk#P Opt B3LYP/gen pseudo=read complex Pd v1 0 2 C 8.89318310 9.90388210 6.72569337 C 9.52931379 8.77525770 6.27102032 H 9.29586123 7.93893890 6.60431879 C 10.52592748 8.89096200 5.30965653 H 10.95942133 8.13380930 4.98695425 C 10.85850598 10.13123090 4.84438728 H 11.51852449 10.22866610 4.19609286 C 10.20972534 11.23549650 5.34144511etc.

etc. H 4.15752044 17.83312399 10.48668123 H 5.63848578 17.14049639 11.10318367

N C O H 06-31G(d)****Pd 0CEP-121G****

Pd 0CEP-121G

ECP example

Recomendations for basis set selection• Always a compromise between accuracy and computational cost!

• With the increase of basis set size, calculated energy will converge (complete basis set (CBS) limit).

• Special cases (anion, transition metal, transtion state)

• Use smaller basis sets for preliminary calculations and for heavy duties (e.g., geometry optimizations), and use larger basis sets to refine calculations.

• Use larger basis sets for critical atoms (e.g., atoms directly involved in bond-breaking/forming), and use smaller basis sets for unimportant atoms (e.g., atoms distant away from active site).

• Use popular and recommended basis sets. They have been tested a lot and shown to be good for certain types of calculations.