1

Numerical methods vs. basis sets

in quantum chemistry

M. Defranceschi

CEA-Saclay

2

Various molecular ab initio models

Minimization problem

where

,1),R(L),R(H;,HinfE 2N32a

N31N

Nji1 ji

N

1i k ki

kN

1ixN

xx

1

xx

zH

i

3

For sake of simplicity

• non relativistic equations,

• time-independent model,

• nuclei are points at fixed known positions,

• real-valued functions,

• spin not explicitly considered,

simplifications to make it more convenient

4

Vector space considered

• Physical functions are square integrable functions (three dimensional measure in the Lebesgue sense)

» Hilbert space

» Reduced to a subspace

»

)R(L N32

)R(L N32a

)R(H N31

5

Overwhelming numerical difficulties

• Problem too difficult to be solved numerically

• Vector space too large

• Non linear terms

)R(L N32a

Nji1 ji xx

1

6

Two classes of simplification

• Rigourous energy/approximated

wavefunction• Hartree-Fock approx.• Restriction to a set of

functions

• Rigourous density/approximated

energy• Density functional

approx.

ijji

N21

where

),...,,det(!N

1

7

Hartree-Fock settings

ijji31

i

2

N

1i

N

1i

2i

2iN1

HF

ijjiNHF

),R(H

;yx

)y,x(

2

1dxdy

yx

)y()x(

2

1

V),...,(E

inf

t;determinan a is ;,HinfI

8

Mathematical fundation

• Define the energy functional E() on a set X of functions (the set of all the possible states of the molecule).

• Then find a function (the ground-state) satisfying some given constraint (i.e. constant number of electrons) and minimize the energy E on the convenient set of states :

X0 )(J 0

)(J,X);(Einf)(EI 0

9

Notion of physical space

• What are the variables of ?• Physical notion : coordinates can be either

position or momentum (or both) or any other quantity.

)R(H 31

3R

3 to1i )p,x(or )p(or )x( iiii

10

First ideas in position space

• Analytical solutions• Numerical solutions

– Radial function of in a one-center approximation

– Spheroidal cooordinates for diatomic molecules– Complete numerical integration for diatomic molecules

In the case of atoms numerical integration are reliable

4n with AHn

11

The quantum chemist procedure

• Molecules are not considered as a whole but as constructed from atoms.

• Use of atomic basis sets• Slater type• Gaussian type• Any functions which contain the correct physical

information.• The procedure most widely used consists in writing the

molecular orbitals as LCAO which belong to a given complete set of the Sobolev space )R(H 31

12

Manageable approximate solutions

• Infinite basis sets are impracticable

• Truncated basis sets

• Large expansions but often to small

• Tendency towards linear dependence

• Inherent deficiencies for GTO– cusp problem– wrong asymptotic decay

13

Some attempts of numerical solutions

• Integration over a numerical grid

• Finite element method

• Momentum space direct numerical integration

• Numerical solution using a wavelet basis

14

Finite element method

• Very accurate results for even time-independent problems for simple systems.

• Large storage requirement for the FE matrices for extended three-dimensional systems

• Removal of the singularities inherent in the nuclear potentials.

15

Momentum space approach(1)

• In position space HF equations are integro-differential

• FT of operators and not of functions

iiiHFh

16

Momentum space approach(2)

• In momentum space HF equations are first order integral equations

)p()qp()q(Wq

dq

2

1

)qp()q(w2)q(Sq

dq

2

1)p(

2

p

ii

2/N

1jj

*ij22

i

2/N

1j

*jj22i

2

)qp()p(dpe)r()r(dr)q(W

and eZ)q(S where

j*i

r.iqj

*iij

N

1A

R.iqA

A

17

Momentum space approach(3)

• The solutions for bound states ( ) can be obtained by an iterative procedure starting with a LCAO in momentum space ( a modified Lanczos procedure).

• Enables to recover basis functions, and then basis sets not limited in size

• Enables to recover the asymptotic behavior.• Removal of the singularities inherent in the

nuclear and interelectronic potentials.

0i

18

Momentum space approach(4)

trial function

first iterate Slater function

1.162569 1.247735 1.248098

0.724396 0.705793 0.750513

1.358986 1.727544 2.322622

2/1

p1

2/12

2p

2/14p

19

Momentum space approach(5)

• Disadvantage of a FT of a function is that all information about its support or its singularities is lost.

• A function with high variations of momenta is hardly interpretable

• A compactly supported function requires a lot of sinusoidal functions

20

A wavelet approach(1)

• The idea is to realize a decomposition with vanishing functions which leads to a momentum representation involving a position parameter

• Functions depending on two variables linked to momentum and position are used

Rb,a ,e)bx(g)x(w

w)x(dx)b,a(C

iaxb,a

*b,a

21

A wavelet approach(2)

• A representation is obtained by means of a decomposition of the Schrödinger operator onto an orthonormal wavelet basis.

• scaling function

• wavelet mother

j,k ),kx2(2)x( j2/jjk

Zj,k ),kx2(2)x( j2/jjk

22

A wavelet approach(3)

• The approach is related to multiresolution analysis, which is a decomposition of the Hilbert space into a chain of closed subspaces.

• The family defined by the scaling function constitutes an o.n. basis set for Vj. Let Wj be the space containing nthe difference in information between Vj-1 and Vj. It allows to decompose

...VVVVV... 21012

jZj

2 W)R(L

23

A wavelet approach (4)

• The two part of a Fock operator has to be treated in two differents ways:– The NS form of the Laplacian operator is

solved iteratively– The NS form of the potential term is obtained

by a quadrature formula

24

A wavelet approach(5)

• The matrix representation of an operator applied to a vector may be depicted

n

n

2

1

1

n

n

2

2

1

1

n

nn

2

22

1

11

s

d...2s

d

s

d

s

d...s

d

s

d

*

G

BA...

G

BAG

BA

25

A wavelet approach (6)

STO-1G STO-2G STO-3G Slater0.4244132

0.4244099

7.78 10-6

0.4857612

0.4857904

6.01 10-5

0.4967535

0.4968063

1.06 10-4

0.5000000

0.5002572

5.14 10-4

-0.8488264

-0.8486610

1.95 10-4

-0.9715744

-0.9711148

4.73 10-4

-0.9937322

-0.9929722

7.65 10-4

-1.000000

-0.9985268

1.47 10-3

-0.4244132

-0.4242511

3.82 10-4

-0.4858132

-0.4853243

1.01 10-3

-0.4969787

-0.4961660

1.64 10-3

-0.500000

-0.4982696

3.46 10-3

error

potd

pot

error

kind

kin

errord

26

Conclusion

• The numerical development is far from the state of the art of the current quantum chemistry practice based on the use of atomic basis sets.