Numerical Integration in DFT

Patrick TamukongThe Kilina Group

Chemistry & Biochemistry, NDSU

The Eigenvalue Problem in DFT

rdrρrvΨVTΨMinrdrρrvρFρE eeρΨ

v

In DFT, we seek ρEMinE v

Nρ0

where the energy functional is

and the electron density is subject to the constraints

Nrdrρ 0rρ

rdρ

221

Due to the analytical complexity of exchange and correlation energy formulas, integrations are performed numerically

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Partitioning of the IntegralExpress the integral as a sum over atomic centers

A

AII

where A A

AAAAAAAA

A rdRrFRrwrdrFrwrdrF

II

Partition or weight function

Function to be integrated

0rw A 1rw

AA 3r R

for any

The partition or weight function fulfills the conditions

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Integral at Atomic CenterEach integral at atomic centers is approximated as a sum of

shell integrals over a series of concentric spheres centered at the nucleus of the atom

The function to be integrated is

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A 0

A2

AAA rdrrF4π

I

where AΩ

AAAA sdsrf4π

1rF

AAA ddθθsinsd

integration over shell of radius Ar Surface element in

spherical coordinates

srRFsrRwsrf AAAAAAA

The Partition FunctionThe partition function or nuclear weight at a given point is

The are hyperbolic coordinates defined as

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where

rz

rp

rp

rprw A

BB

AA

AB

ABA rμsrp is the unnormalized cell

function of atom A, composed of independent pair contributions rμs AB

rμAB

AB

BA

AB

BA

AB R

rr

R

RrRrrμ

1Rμ AAB 1Rμ BAB

3AB r1rμ R

The Cell FunctionIt must be close to unity near nucleus A and close to zero

near other nuclei, thus the contribution between atoms A and B, , decreases monotonically as follows

is subject to the conditions

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rμs AB

1μ1μs0 11s 01s 0dμ

ds 0

dμ

ds

1μ

Gräfenstein, J.; Cremer, D. J. Chem. Phys. 2007, 127, 164113.

Becke’s Definition of The Cell FunctionAccording to Becke

Becke found k = 3 to be the optimum value for a sufficiently well-behaved . Since , it follows that

7Becke, A. D. J. Chem. Phys. 1988, 88, 2547.

μp12

1μs k

μppμp k1k

Where the polynormials are such that μpk

31 μ

2

1μ

2

3μp

rμs AB

μpμ-p 33

μs1μp12

1μs 3

Properties of the Hyperbolic CoordinatesConsider

Using the cosine rule

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Tamukong, P. K. Extension and Applications of GVVPT2 to the Study of Transition Metals. Ph.D. Dissertation, University of North Dakota, Grand Forks, ND, 2014 (http://gradworks.umi.com/36/40/3640951.html).

1R

rrrμ1

AB

BAAB

ABAAABBABA

AAABAB2A

2

ABA2B

Rcos2rRrrrr

cos2rRRrRrr

BA

ABA

BA

ABAA

AB

BAAB rr

R2r

rr

Rcos2r

R

rrrμ

Properties of the Hyperbolic CoordinatesThus only within a sphere of radius

At a fixed radius of a given sphere around atom A, and are even functions of the angle

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Tamukong, P. K. Extension and Applications of GVVPT2 to the Study of Transition Metals. Ph.D. Dissertation, University of North Dakota, Grand Forks, ND, 2014 (http://gradworks.umi.com/36/40/3640951.html).

0rμAB

ABA R2

1r

that is 0rμAB

iff ABA R2

1Rr

Ar Br

rμ AB

A

From BA

ABA

BA

ABAA

AB

BAAB rr

R2r

rr

Rcos2r

R

rrrμ

AB

ABA

A

B sinr

Rr

d

dr

AB

ABA2A

B2

cosr

Rr

d

rd

ABA

A

A

AB sinrr

2r

d

dμ

ABA

A2A

AB2

cosrr

2r

d

μd

Properties of the Hyperbolic CoordinatesThus has its maximum at and minimum

at

Meanwhile has its maximum on the sphere

10

when

ABr πA

0A

AABμ

ABAA R

2

1rs and minimum when πA

0A

Alternative Definition of Cell FunctionStratmann et al. alternatively define the cell function as

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Where is a piece-wise odd function defined as

Stratmann, R. E.; Scuseria, G. E.; Frisch, M. J. Chem. Phys. Lett. 1996, 257, 213.

μg12

1μs a

μga

aμ1

aa,μμz

aμ1

μg aa 1a0

753

a a

μ5

a

μ21

a

μ35

a

μ35

16

1μz

Alternative Definition of Cell FunctionWithin the limits , the function is

subject to the constraints

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aa,μ μza

μzμz aa 0dμ

dza 1aza 1aza 0dμ

dz

aμ

a

The function has zero second and third order derivatives at and leads to

μza

aμ 2

10s

32a

35

dμ

ds

0μ

The Stratmann et al. cell function satisfies

aμif0

aμif1μs 80.64814814

272

35a

reliably

from the requirement that the derivatives of the Becke and Stratmann cell functions coincide at 0μ

Selection of Significant FunctionsIn performing integrations, advantage is taken of the fast

decaying nature of Gaussian atomic orbitals such that for each grid point, only such functions that are numerically significant (according to a user-specified criterion, ) are considered

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For grid point , a set of significant functions is chosen which satisfies

radius of considered sphere

ε

εiχ Aμ

gr

gs Aμχ

gAμ sχ ελRr A

μAg

Selection of Significant FunctionsTo maximize computational efficiency, blocks of grid

points are used, e.g., a sphere of grid points with a set of significant basis functions

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Gg

gG sS

ggrG

GAμ Sχ if ελRr A

μAg

for some Grg

AA rGAγ Lrχ

AA rGAυ Lrχ

Thank You

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