SCF methods, basis sets, and integrals

Lecture IV: Integrals

Wim Klopper

Abteilung fr Theoretische Chemie, Institut fr Physikalische ChemieKarlsruher Institut fr Technologie (KIT)

ESQC 2011, Torre Normanna, 9/18 10/01/2011

The Gaussian-product theorem

The great success of GTOs isbased on the fact that allnecessary integrals are easilyevaluated analytically.

The most important reason forthis efficiency is the Gaussian-product theorem (GPT).

0

0.2

0.4

0.6

0.8

1

-3 -2 -1 0 1 2 3

f(x)

x

The figure shows the product of the (unnormalised) s-type GTOsa with exponent = 0.25 at A = (1, 0, 0) and b with exponent = 0.50 at B = (1, 0, 0) ,

a(r) = e(rA)2 , b(r) = e

(rB)2

The two-centre product a(r)b(r) is again a Gaussianp centred at the centre of gravity P.

The Gaussian-product theorem

The product of the GTOs a and b can be written as

a(r)b(r) = e + (AB)2e(+)(rP)

2

with

P =A + B

+

The factor exp( + (AB)2) isknown as pre-exponential factor.Obviously, this factor vanishes forlarge distances between A and B .

0.1

1

10

100

1000

0.001 0.01 0.1 1 10 100 1000

Dis

tanc

e

Exponent

Note that for two like exponents of the order of 103 a20 , thedistance |AB| must be 100 a0 to make the pre-exponentialfactor smaller than 106.

The overlap integral Using the GPT, We can easily compute the overlap integral Sab

between two (real, unnormalised) s-type Gaussians,a(r)b(r)dr = e

+ (AB)2e(+)(rP)

2

dr

= e

+ (AB)2e(+)r

2

dr

= e

+ (AB)2e(+)x

2

dx

e(+)y

2

dy

e(+)z

2

dz

= e

+ (AB)2(

+

) 32

This equation reveals another important property of integralsover Gaussians: the 3D integral factorises into a productof three 1D integrals.

Normalisation constant: Na =(2

) 34 .

The overlap integral

The figure shows the overlapintegral Sab for two s-typeGaussians with exponent = = 1 a20 as a function ofthe distance |AB| (solid line).

The dashed lines are overlapintegrals with exponents 10 timeslarger and 10 times smaller.

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

S(A

B)

AB

For the integral over contracted Gaussians (CGTOs), the overlapintegral becomes

S =

n

a=1

n

b=1

caSabcb

In general, a large number of integtals over primitive functionscontribute to a smaller number of integrals over CGTOs.

Primitive Cartesian GTOs The primitive Cartesian GTO is

a(r) = xiAy

jAz

kA exp(r2A), rA = rA

Integrals over real-valued spherical-harmonic GTOs

a(r) = Slm(xA, yA, zA) exp(r2A)

(where Slm(xA, yA, zA) is a real solid harmonic), can beobtained by transforming the integrals over primitive CartesianGTOs with a corresponding transformation matrix.

S = CT SC

Usually, this transformation is done after the contraction:

primitive Cartesian GTOs contracted Cartesian GTOs contracted spherical-harmonic GTOs

Primitive Cartesian GTOs As already mentioned, the factorisation of the Cartesian GTOs is

an important property,

a ={xiA exp(x2A)

}{yiA exp(y2A)

}{ziA exp(z2A)

}

We can thus focus our attention on only one Cartesiancomponent, say x,

Gi(x, ,Ax) = xiA exp(x2A)

The self overlap of the x component is

Gi|Gi =(2i 1)!!

(4)i

2, G0|G0 =

2

All we have used thus far is the definite integral

x2neax

2

dx =1 3 5 . . . (2n 1)

(2a)n

a

Recurrence relations for Cartesian GTOs

The differentiation property of Cartesian GTOs is needed onseveral occasions,

GiAx

= Gix

= 2Gi+1 iGi1

In words, differentation of a dxy-type GTO with respect to x givesa linear combination of py and fx2y, etc.

For higher derivatives, we obtain

n+1Gi

An+1x=

(

Ax

)n(2Gi+1 iGi1) = 2

nGi+1Anx

i nGi1Anx

We thus find (besides Gi+1 = xAGi):

Gn+1i = 2Gni+1 iGni1, with Gni =

nGiAnx

Gaussian overlap distributions We define the Gaussian overlap distribution

ab(r) = a(r)b(r)

which factorises as

ab(r) = Gi(x, ,Ax)Gj(x, ,Bx) = xij(x, , ,Ax, Bx) . . .

Due to the GPT, xij may be written as

xij = Kxabx

iAx

jB exp(x2P), with = +

Kxab is the x component of the pre-exponential factor.

In the above equation, we give x relative to Ax, xB and xP. Wetherefore rewrite the equation using

xA = xAx = x Px + (Px Ax) = x Px +XPA = xP +XPAxB = xBx = x Px + (Px Bx) = xP +XPB

Properties of overlap distributions

We have the obvious relationships

xAxij =

xi+1,j , xB

xij =

xi,j+1, XAB

xij =

xi,j+1 xi+1,j

Differentiating the overlap distributions yieldsxijAx

= 2xi+1,j ixi1,j ,xijBx

= 2xi,j+1 j xi,j1

We furthermore note that

XPA = Px Ax =Ax + Bx

+

Ax =

(Bx Ax) =

XAB

XPB = Px Bx =Ax + Bx

+

Bx =

(Ax Bx) =

XAB

It is sometimes convenient to work with Px and XAB inthe place of Ax and Bx.

The ObaraSaika scheme for Sij We consider the integral

Sij =

xijdx

This integral is invariant to a translation of the coordinate systemalong the x-axis,

SijAx

+SijBx

= 0

This yields the translational recurrence relation

2Si+1,j i Si1,j + 2Si,j+1 j Si,j1 = 0

This recurrence relation alone is not useful, because there aretwo terms with quantum number i+ j + 1.

The idea is to first compute S00 and then to obtain allother integrals from the recurrence relation.

The ObaraSaika scheme for Sij In order to be useful, the translational recurrence relation

2Si+1,j i Si1,j + 2Si,j+1 j Si,j1 = 0must be combined with the horizontal recurrence relation,

Si,j+1 Si+1,j = XABSij By doing this, we obtain the ObaraSaika (OS) recurrence

relations for the Cartesian overlap integrals,

Si+1,j = XPASij +1

2(i Si1,j + j Si,j1)

Si,j+1 = XPBSij +1

2(i Si1,j + j Si,j1)

We start with S00 = Kxab

and then compute

S10 = XPAS00

S20 = XPAS10 +1

2S00, etc.

The ObaraSaika scheme for Sij

The fundamental ideas are:1. Translational invariance:

Sij/Ax + Sij/Bx = 0

2. Horizontal recurrrence:xB xA = XAB

!

S00

!

S10

!

S01

!

S11

!

S02

!

S20

!

S21

!

S12

!

S30

!

S03

Each Sij in the triangle is compute from one of the two above it andfrom the two above that one.

The target integral Sij may be generated in many different ways. Note that horizontal recurrrence relation can be applied to transfer

quantum numbers from i to j and vice versa for all kinds of basisfunctions, also contracted Gaussians.

xA| = |xB XAB|

ObaraSaika for multipole moments The ObaraSaika scheme may be applied to multipole-moment

schemes in a slightly modified form,

Sefgab = a|xeC yfC z

gC|b = SeijSfklSgmn

The x component is

Seij = Gi|xeC|Gj =

xeC

xijdx

Translational invariance for this integral means thatSeijAx

+SeijBx

+SeijCx

= 0

Furthermore, the horizontal recurrence relation for the order ofthe multipole operator is (xC = xA +XAC, etc.)

Se+1ij = Sei+1,j +XACS

eij = S

ei,j+1 +XBCS

eij

ObaraSaika for multipole moments Putting it all together yields

Sei+1,j = XPASeij +

1

2

(i Sei1,j + j S

ei,j1 + eS

e1ij

)

Sei,j+1 = XPBSeij +

1

2

(i Sei1,j + j S

ei,j1 + eS

e1ij

)

Se+1ij = XPCSeij +

1

2

(i Sei1,j + j S

ei,j1 + eS

e1ij

)

These recurrence relations may be used in conjunction with thehorizontal recurrences

Se+1ij = Sei+1,j +XACS

eij = S

ei,j+1 +XBCS

eij

andSei,j+1 = XABS

eij + S

ei+1,j

The final integrals are obtained by multiplying the x, y and zfactors, followed by a transformation to contractedspherical-harmonic components.

Differential operators

We now consider the integrals in a slightly modified form,

Defgab = a|e

xef

yfg

zg|b = DeijDfklDgmn

The x component is

Deij = Gi|e

xe|Gj =

GieGjxe

dx

The trick we use here is that we can differentiate the GaussianGi(x, ,Ax) with respect to the electron coordinate x or thebasis-function centre Ax, because the function depends on thedifference (xAx)

Gi(x, ,Ax)

x= Gi(x, ,Ax)

Ax

Differential operators

Since Gi/x = Gi/Ax and Sij/Ax = Sij/Bx, weobtain

Deij = eSij/A

ex = (1)eeSij/Bex

Furthermore, since XPA/Ax = / and XPB/Ax = /,we obtain the ObaraSaika recurrence relations

Dei+1,j = XPADeij +

1

2

(iDei1,j + j D

ei,j12eDe1ij

)

Dei,j+1 = XPBDeij +

1

2

(iDei1,j + j D

ei,j1+2eD

e1ij

)

De+1ij = 2Dei+1,j iDi1,j

The horizontal recurrence relation becomes

Dei,j+1 Dei+1,j = XABDeij + eDe1ij

Momentum and kinetic-energy integrals Consider the one-electron integrals

Pab = iGa||Gb (linear momentum)Lab = iGa|r|Gb (angular momentum)

Tab = 1

2Ga||Gb (kinetic energy)

The z components of the momentum integrals, for example, maybe easily computed from

P zab = iSijSklD1mnLzab = iGa|x

y xy|Gb = i

(S1ijD

1klSmn D1ijS1klSmn

)

For the kinetic-energy integral, we obtain

Tab = 1

2

(D2ijSklSmn + SijD

2klSmn + SijSklD

2mn

)

Coulomb integrals over spherical Gaussians

We consider the electrostatics of the (normalised) sphericalGaussian charge distributions

p(r) =(

)3/2exp(r2P), q(r) =

(

)3/2exp(r2Q)

The normalisation means thatp(r)dr =

q(r)dr = 1

The electrostatic potential at C due to p is

Vp(C) =

p(r)

rCdr

(0,0,0)

C

r rC = | r ! rC |

!

" p r( )

Coulomb integrals over spherical Gaussians

The energy of repulsion between the charge distributions pand q is

Vpq =

p(r)q(r

)|r r| drdr

The difficulty with this integral and Vp(C) is that they do notfactorise into products of x, y and z components due to thedistances (square roots) rC and |r r|.

Integrals over rk with k even are easy, those with k odd aredifficult.

However, the integrals factorise again after the integral transform

1

rC=

1

exp(r2C t2)dt

This is the key step in treating Coulomb integrals.

Coulomb integrals over spherical Gaussians

In the integral Vp(C), we have a product of two Gaussians:p(r) and exp(r2C t2).

This product yields a new Gaussian centred at

S = (P + t2C)/( + t2)

according to the Gaussian-product theorem (GPT),

Vp(C) =3/2

2

{exp[( + t2)r2S]dr

}exp

( t

2

+ t2R2PC

)dt

The spatial integral can easily be computed and we obtain

Vp(C) =23/2

0

( + t2)3/2 exp

(R2PC

t2

+ t2

)dt

which can be solved after substituting u2 = t2/( + t2).

The Boys function Since dt = (1 u2)3/2du, we obtain

Vp(C) =

4

1

0

exp(R2PCu2)du

The integration over all space (x, y and z from to) hasbeen replaced by a one-dimensional integration over a finiteinterval.

This integral is the Boys function Fm(x) with m = 0,

Fm(x) =

1

0

t2m exp(xt2)dt, F0(x) =

4xerf(x)

(erf is the error function).

We can thus write

Vp(C) =

4

F0(R

2PC)

Calculating the Boys function

The Boys function can be computed efficiently by pretabulatingFm(xk) for a series of grid points xk.

For example, we can tabulate Fm(xk) for m = 0, 1, 2, . . . ,mmax + 5 at regular intervals x1 = 0.0, x2 = 0.1, x3 = 0.2, . . . ,xn = 2mmax + 36.

The Boys functions Fm(x) may then be computed with machineprecision from a six-term Taylor expansion around xk,

Fm(x) = Fm(xk + x) = Fm(xk) Fm+1(xk)x+ 12Fm+2(xk)(x)2 16Fm+3(xk)(x)3 + 124Fm+4(xk)(x)4 . . .

x4 x

5 x

6 x

7 x

8 x

9

F7

F6

F5

F4

F3

F2

F1

F0

x

!x

The figure shows thegrid points involved incomputing F2(x) withx7 < x < x8.

Calculating the Boys function We note in passing that the exponential exp(x) can be

computed similarly by pretabulating exp(xk) at a number ofgrid points xk.

This number can be chosen such that a four-term Taylorexpansion is enough to obtain machine precision.

At the grid points xk, the Boys functions are computed bydownward recursion,

Fm(x) =2xFm+1(x) + exp(x)

2m+ 1, F(x) = 0

Fm(x) can be set equal to zero for sufficiently large m. For large x, we have

Fm(x) (2m 1)!!

2m+1

x2m+1, (x large)

The OS scheme for one-electron Coulomb integrals

We write the Coulomb integral as

0ijklmn = Ga|1

rC|Gb =

ab(r)

rCdr =

xij

ykl

zmn

rCdr

We obtain 0ijklmn from N000000 = 2 Kxyzab FN (R

2PC) and

Ni+1,jklmn = XPANijklmn +

1

2(iNi1,jklmn + j

Ni,j1,klmn)

XPCN+1ijklmn 1

2(iN+1i1,jklmn + j

N+1i,j1,klmn)

Ni,j+1,klmn = XPBNijklmn +

1

2(iNi1,jklmn + j

Ni,j1,klmn)

XPCN+1ijklmn 1

2(iN+1i1,jklmn + j

N+1i,j1,klmn)

and similarly for y and z.

The McMurchieDavidson scheme Before we turn our attention to the two-electron repulsion

integrals, we note that the following three integral-evaluationtechniques are currently in use:

1. The ObaraSaika scheme2. The McMurchieDavidson scheme3. Rys quadrature

The idea of the McMurchieDavidson scheme is to expand theoverlap distribution ab in Hermite Gaussians,

xij =

i+j

t=0

Eijt t, t = (/Px)t exp(x2P)

and similarly for ykl and zmn.

In the McMurchieDavidson (MD) scheme, integrals overHermite Gaussians are evaluated and transformedto the Cartesian Gaussian basis using the expansioncoefficients Eijt .

The MD expansion coefficients

In order to compute the expansion coefficients Eijt , we considerthe incremented distribution

xi+1,j =

i+j+1

t=0

Ei+1,jt t

Of course, xi+1,j = xAxij = xPxij +XPAxij , and furthermore

xPt = tt1 +1

2t+1

We thus obtain

xPxij =

i+j

t=0

Eijt (tt1+1

2t+1) =

i+j+1

t=0

{(t+ 1)Eijt+1 +

1

2Eijt1

}t

Here, we assume that Eijt = 0 when t < 0 or t > i+ j.

The MD expansion coefficients We have established that

xi+1,j =

i+j+1

t=0

Ei+1,jt t

and

xi+1,j =

i+j+1

t=0

{(t+ 1)Eijt+1 +

1

2Eijt1 +XPAE

ijt

}t

We therefore arrive at the following McMurchieDavidsonrecurrence relations for the expansion coefficients:

Ei+1,jt =1

2Eijt1 +XPAE

ijt + (t+ 1)E

ijt+1

Ei,j+1t =1

2Eijt1 +XPBE

ijt + (t+ 1)E

ijt+1

E000 = Kxab

The McMurchieDavidson scheme For all three Cartesian coordinates, we have

tuv = tuv =

(

Px

)t(

Py

)u(

Pz

)vexp(r2P)

and

ab =

i+j

t=0

k+l

u=0

m+n

v=0

Eijt Eklu E

mnv tuv

Thus, using the MD scheme, the Coulomb integrals becomes

0ijklmn =

i+j

t=0

k+l

u=0

m+n

v=0

Eijt Eklu E

mnv

tuvrC

dr

Furthermore,

tuvrC

dr =2

(

Px

)t(

Py

)u(

Pz

)vF0(R

2PC)

The McMurchieDavidson recurrence relations The Coulomb integrals are written as

0ijklmn =

i+j

t=0

k+l

u=0

m+n

v=0

Eijt Eklu E

mnv

tuvrC

dr

=2

i+j

t=0

k+l

u=0

m+n

v=0

Eijt Eklu E

mnv R

0tuv

Here, we have introduced the auxiliary integrals

RN000 = (2)NFN (R2PC)

The integrals R0tuv are obtained from the recurrence relations

RNt+1,uv = tRN+1t1,uv +XPCR

N+1tuv

RNt,u+1,v = tRN+1t,u1,v + YPCR

N+1tuv

RNtv,u+1 = tRN+1tu,v1 + ZPCR

N+1tuv

GauRys quadrature We have seen that the Coulomb integrals are obtained as a

linear combination of Boys functions

0ijklmn =M

N=0

cNFN (R2PC), M = i+ j + k + l +m+ n

where the coefficients cN depend on the exponents andcoordinates involved.

Since FN (x) = 10t2N exp(xt2)dt, we may write

0ijklmn =

1

0

pM (t2) exp(R2PCt2)dt

pM (t2) is a polynomial in t2 of degree M . The integral can beevaluated by means of a GauRys quadrature usingL =

[M2

]+ 1 quadrature points (roots t and weights w),

0ijklmn =L

=1

w pM (t2) exp(R2PCt2)

Two-electron Coulomb integrals

We now turn our attention to the two-electron integral Vpq, whichis the energy of repulsion between the (normalised) chargedistributions p and q,

Vpq =

p(r)q(r

)|r r| drdr

Integration over r yields

Vpq =

4

F0(r

2Q)p(r)dr

The charge distribution p(r) is a Gaussian, F0(r2Q) is anintegral over a Gaussian, and their product is again (an integralover) a Gaussian by virtue of the GPT.

The integration over r is easy and the integration over t[hidden in F0(r2Q)] remains.

Two-electron Coulomb integrals We obtain

Vpq =

4

1

0

(t2 + )3/2exp

(t2R2PQt2 +

)dt

This integral can be written as

Vpq =

4

F0(R

2PQ), =

+

which can be verified by substituting u2 = +t2+ t2.

is the reduced exponent.

Recurrence relations for Cartesian Gaussians other than s-typefunctions may be obtained in a mannner similar to theone-electron Coulomb integrals, e.g., with the auxiliary functions

N0000;0000;0000 =25/2

+

Kxyzab Kxyzcd FN (R

2PQ)

Two-electron Coulomb integrals

The two-electron Coulomb integrals are

ab|r112 |cd = gabcd = 0ijij;klkl;mnmn

Since there are four Gaussians involved (Ga, Gb, Gc and Gd) thetotal number of integrals scales as N4, where N is the size ofthe basis set.

When we increase the basis set in a series of calculations of thesame small molecule, there is little we can do about the O(N4)scaling.

When we run calculations on a series of molecules of differentsize (e.g., on the alkanes CNH2N+2) in a given Gaussian basis,then many two-electron Coulomb integrals are very small andcan be ignored.

In that case, the number of significant integrals scalesas O(N2).

Scaling of two-electron Coulomb integrals To facilitate the discussion, we consider two-electron Coulomb

integrals over (unnormalized) s-type Gaussians,

gabcd = ab|r112 |cd = 00000;0000;0000 =25/2

+

Kxyzab Kxyzcd F0(R

2PQ)

We note that25/2

+

=

4

(

)3/2(

)3/2and Sab =

(

)3/2Kxyzab

The two-electron Coulomb integral can thus be written as

gabcd =

4

SabScdF0(R

2PQ)

We furthermore know that

F0(x) 1 and F0(x) 1

2

x

Scaling of two-electron Coulomb integrals

We find that the Coulomb integral is bounded by

gabcd SabScd min(

4

,

1

RPQ

)

For large molecular systems, the number of nonzero overlapintegrals Sab scales as O(N).

When the number of significant overlap integrals begins toincrease as N , the number of significant two-electron Coulombintegrals will begin to increase as N2.

Note that R1PQ will not be smaller than 106 in any practicalcalculation.

The number of significant two-electron Coulomb integrals willdepend at least quadratically on the size of the system.

Prescreening of integrals In large systems, the number of significant two-electron integrals

increases only quadratically.

In order to exploit this fact, we need a strict upper bound for themagnitude of the integrals.

Such a bound is provided by the Schwarz inequality,

|(ab|cd)| QabQcd

with Qab =

(ab|ab) and Qcd =

(cd|cd). Before an integral is computed, the product QabQcd is compared

with some threshold . The integral is only computed if

QabQcd

Typical values are = 107 108 for small molecules. The upper bound is also useful when we ask how the

integral contributes to the energy or the Fock matrix.

Prescreening of integrals In closed-shell HartreeFock theory, the Fock matrix is

F = h +

D[2(|) (|)]

Using real basis functions, we have the following permutationalsymmetry among the two-electron integrals:

(|) = (|) = (|) = (|)= (|) = (|) = (|) = (|)

Hence, we compute only ca. 1/8 of the total number of integralsand use each integral in various manners,

F = F + 4D(|)F = F + 4D(|)F = F D(|)F = F D(|)F = F D(|)F = F D(|)

Prescreening of integrals

The density matrix elements are known when the integrals areevaluated. They can be incorporated in the prescreening tests.

The evaluation of the integral (|) is only needed if

QQDmax

where

Dmax = max {4|D |, 4|D|, |D|, |D|, |D|, |D|}

Concerning the HartreeFock energy, we could screen with

QQmax {4|DD|, |DD|, |DD|}

but this leaves an unmonitored error in the Fock matrix.

This last screening is however useful for energy-relatedproperties such as the nuclear forces.

The direct SCF procedure In conventional SCF procedures, the integrals are computed only

once and stored on disk for later use. In such procedures, theSchwarz screening helps to eliminate integrals once and for all.

In direct SCF procedures, the integrals are re-evaluated in eachSCF iteration.

The prescreening is then performed in conjunction with thedensity matrix, which is usually done for batches of integrals,

DMN = max |D | M, N

QMN = max |Q | M, N

Furthermore, important savings in the number of calculatedintegrals may be obtained by considering the change of the Fockmatrix in two consecutive iterations.

The screening can then be performed using the change of thedensity matrix,

D(i) = D(i) D(i1)

The direct SCF procedure

In the direct SCF procedure, the Fock matrix in iterationnumber i is computed from

F (i) = F(i1) +

D(i) [2(|) (|)]

Only those integrals are required which are related to significantchanges in the density matrix from one iteration to the next.

Close to convergence, screening with D(i) is extremelyefficient.

Furthermore, if the Coulomb and exchange matrices areconstructed separately, we have the following two screeningcriteria:

QQ {4|D |, 4|D|} QQ {|D|, |D|, |D|, |D|}

The RI approximation

The idea of the resolution-of-the-identity (RI) approximation is toavoid four-index (or four-centre) two-electron integrals.

In a naive approach, we can insert an approximation to the unityoperator represented in an orthonormal auxiliary basis {P },

1

P

|P P |

We then obtain, for example,

(|)

P

(|P )P, P =P (1)(1)(1)dr1

(|P ) is a three-index two-electron repulsion integral,

(|P ) =

(1)(1)r112 P (2)dr1dr2

RI approximation with non-orthonormal basis

If we approximate the unity operator by a projection operatoronto a non-orthonormal auxiliary basis {P }, then we have

1

P,Q

|P S1PQQ|

Note that S1PQ is a matrix element of the inverse overlap matrix,

S1PQ (S1

)PQ

We obtain(|)

PQ

(|P )S1PQQ

It has turned out that the most accurate RI approximation isobtained by using the Coulomb metric

RI approximation with Coulomb metric

Using the Coulomb metric, we approximate the unity operator asfollows:

1

P,Q

|P (P |Q)1(Q|,

where(Q| =

Q(2)r

112 dr2

Note that (P |Q)1 is a matrix element of the inverse of thetwo-index Coulomb integrals,

S1PQ (C1

)PQ

, CPQ =

P (1)r

112 Q(2)dr1dr2

We obtain

(|)

PQ

(|P )(P |Q)1(Q|)

RI approximation with Coulomb metric Alternatively, we expand the orbital product in a basis{P },

=

P

cP P

We then minimise the self-repulsion of the error,

( | ) = min

This immediately leads to the set of linear equations

Q

(P |Q)cQ = (P |) P cP =

Q

(P |Q)1(Q|)

Hence,

(|)

P

(|P )cP =

PQ

(|P )(P |Q)1(Q|)

Robust fitting

The RI approximation can be inserted in such a manner, that theerror in the target integral is only quadratic in the error of the fit,

(|)

P

cP (P |) +

Q

(|Q)cQ

PQ

cP (P |Q)cQ

For Coulomb integrals, as before, this leads to

(|)

PQ

(|P )(P |Q)1(Q|)

The robust fitting is, however, important for other two-electronintegrals, for example those over the operator f(r12),

(||)

P

cP (P ||) +

Q

(||Q)cQ

PQ

cP (P ||Q)cQ

where (. . . || . . . ) indicates the integral over f(r12).

Various applications of the RI approximation

The RI approximation is for example used to accelerate thecalculation of the Coulomb operator J (RI-J approximation),especially in DFT.

The RI approximation can also be used to accelerate thecalculation of the exchange operator K (RI-JK approximation) inHartreeFock theory or DFT with hybrid functional (e.g., B3LYP).

Finally, it can also be used to approximate integrals of the type(IA|JB), where I, J are occupied HartreeFock orbitals andA,B are virtual HartreeFock orbitals. These integrals occur inthe MP2 and CC2 theories.

Clearly, different basis sets are needed to approximate orbitalproducts of the types II (RI-J), I (RI-JK) and IA(cbas in Turbomole).

These three types of auxiliary basis sets have beendesigned and optimised for the Turbomole basis sets.

RI-J approximation In the RI-J approximation, the electron density is expanded in an

appropriate auxiliary basis,

J = (|)

P

(|P )cP =

P

(|P )cP D

=

PQ

(|P )(P |Q)1(Q|)D

The formal scaling is no longer N4 but rather N3, assuming thatthe auxiliary basis increases linearly with system size.

It is possible to obtain accurate results (with an error of ca. 0.1mEh per atom) with auxiliary basis sets that are about threetimes larger than the orbital basis.

Asymptotically, the construction of the Coulomb matrix will scaleas N2, as before, but with a much smaller prefactor.

The RI-J approximation can be combined with multipolemethods (MARI-J : multipole-accelerated RI-J).

RI-JK approximation

In the RI-JK approximation, the RI approximation is not onlyused for the Coulomb matrix but also for the exchange matrix,

K =

D(|)

PQ

(|P )(P |Q)1(Q|)D

The formal scaling is still N4, as before. The RI-JK approximation is very useful when relatively large

basis sets are used, for example when the HartreeFockcalculation is followed by a post-HartreeFock treatment.

Similar algorithms have been developed for the RI-MP2 andRI-CC2 methods. The formal scaling is still N5 for thesemethods, as before, but with a much smaller prefactor.

The RI-MP2-F12 methods utilises robust fitting forintegrals over other operators than 1/r12.