# Basis set extrapolation

Basis set extrapolation can be carried out for correlation consistent basis sets using

EXTRAPOLATE,BASIS=basislist,options

where basislist is a list of at least two basis sets separated by colons, e.g. AVTZ:AVQZ:AV5Z. Some extrapolation types need three or more basis sets, others only two. The default is to use $n^{-3}$ extrapolation of the correlation energies, and in this case two subsequent basis sets and the corresponding energies are needed. The default is not to extrapolate the reference (HF) energies; the value obtained with the largest basis set is taken as reference energy for the CBS estimate. However, extrapolation of the reference is also possible by specifying the METHOD_R option.

The simplest way to perform extraplations for standard methods like MP2 or CCSD(T) is to use, e.g.

***,H2O
memory,32,m
gthresh,energy=1.d-8

r =   0.9572 ang, theta =  104.52
geometry={O;
H1,O,r;
H2,O,r,H1,theta}

basis=avtz
hf
ccsd(t)
extrapolate,basis=avqz:av5z

table,basissets,energr,energy-energr,energy
head,basis,ehf,ecorr,etot

This will perform the first calculation with AVTZ basis, and then compute the estimated basis set limit using the AVQZ and AV5Z basis sets. The correlation energy obtained in the calculation that is performed immediately before the extrapolate command will be extrapolated (in this case the CCSD(T) energy), and the necessary sequence of calculations [here HF;CCSD(T)] will be automatically carried out.

The resulting energies are returned in variables ENERGR (reference energies), ENERGY (total energies), and ENERGD (Davidson corrected energy if available); the corresponding basis sets are returned in variable BASISSETS. The results can be printed, e.g., in a table as shown above, or used otherwise. The above input produces the table

 BASIS       EHF           ECORR          ETOT
AVQZ    -76.06600082   -0.29758099   -76.36358181
AV5Z    -76.06732050   -0.30297495   -76.37029545
CBS     -76.06732050   -0.30863418   -76.37595468

The extrapolated total energy is also returned in variable ECBS (ECBSD for Davidson corrected energy if available).

In order to extrapolate the HF energy as well (using exponential extrapolation), three energies are needed. One can modify the input as follows:

extrapolate,basis=avtz:avqz:av5z,method_r=ex1,npc=2

method_r determines the method for extrapolating the reference energy (in this case a single exponential); npc=2 means that only the last two energies should be used to extrapolate the correlation energy (by default, a least square fit to all given energies is used). This yields

 BASIS           EREF            ECORR           ETOT
AVTZ        -76.06061330     -0.28167606    -76.34228936
AVQZ        -76.06600082     -0.29758099    -76.36358180
AV5Z        -76.06732050     -0.30297495    -76.37029545
CBS         -76.06774863     -0.30863419    -76.37638283

Rather than using the default procedure as above, one can also specify a procedure used to carry out the energy calculation, e.g.

extrapolate,basis=avtz:avqz:av5z,proc=runccsd, method_r=ex1,npc=2}

procedure runccsd
hf
ccsd(t)
endproc

Alternatively, the energies can be provided via variables EREF, ECORR, ETOT etc. These must be vectors, holding as many values as basis sets are given.

The possible options and extrapolation methods are:

• BASIS=basissets Specify as set of correlation consistent basis sets, separated by colons.
• PROC=procname Specify a procedure to run the energy calculations
• STARTCMD=command Start command for the energy calculations: the sequence of commands from STARTCMD and the current EXTRAPOLATE is run. STARTCMD must come before the extrapolate command in the input.
• METHOD=key Specifies a keyword to define the extrapolation function, see section extrapolation functionals.
• METHOD_C=key Specifies a keyword to define the extrapolation function for the correlation energy, see section extrapolation functionals.
• METHOD_R=key Specifies a keyword to define the extrapolation function for the reference energy, see section extrapolation functionals.
• VARIABLE=name Specifies a variable name; this variable should contain the energies to be extrapolated.
• ETOT=variable Provide the total energies in variable (a vector with the same number of energies as basis sets are given) If only ETOT but not EREF is given, the total energy is extrapolated.
• EREF=variable Provide the reference energies to be extrapolated in variable (a vector with the same number of energies as basis sets are given)
• ECORR=variable Provide the correlation energies to be extrapolated in variable (a vector with the same number of energies as basis sets are given)
• ECORRD=variable Provide the Davidson corrected correlation energies to be extrapolated in variable (a vector with the same number of energies as basis sets are given). If both ECORR and ECORRD are given, both will be extrapolated.
• MINB=number First basis set to be used for extrapolation (default 1)
• MAXB=number Last basis set to be used for extrapolation (default number of basis sets)
• NPR=number If given, the last NPR values are used for extrapolating the reference energy. NPR must be smaller or equal to the number of basis sets.
• NPC=number If given, the last NPC values are used for extrapolating the reference energy. NPC must be smaller or equal to the number of basis sets.
• XR=array Provide a vector of exponents to be used for defining the extrapolation functional for the reference energy when using the LX functional.
• XC=array Provide a vector of exponents to be used for defining the extrapolation functional for the correlation energy when using the LX functional.
• PR=array Provide the constant $p$ to be used for defining the extrapolation functional for the reference energy.
• PC=array Provide the constant $p$ to be used for defining the extrapolation functional for the correlation energy.

The extrapolation functional is chosen by a keyword with the METHOD, METHOD_R, and/or METHOD_C options. The default functional is L3. In the following, $n$ is the cardinal number of the basis set (e.g., 2 for VDZ, 3 for VTZ etc), and $x$ is an arbitrary number. $p$ is a constant given either by the PR or PC options (default $p=0$). X is a number or a vector given either by the XR or XC options (only for LX; $nx$ is the number of elements provided in X). $A$, $B$, $A_i$ are the fitting coefficients that are optioized by least-squares fits.

• L$x$ $E_{n} = E_{\tt CBS} + A \cdot (n+p)^{-x}$
• LH$x$ $E_{n} = E_{\tt CBS} + A \cdot (n+\frac{1}{2})^{-x}$
• LX $E_{n} = E_{\tt CBS} + \sum_{i=1}^{nx} A_i \cdot (n+p)^{-x(i)}$
• EX1 $E_{n} =E_{\tt CBS}+A\cdot \exp(-C\cdot n)$
• EX2 $E_{n} =E_{\tt CBS}+A\cdot \exp(-(n-1))+B\cdot\exp(-(n-1)^2)$
• KM Two-point formula for extrapolating the HF reference energy, as proposed by A. Karton and J. M. L. Martin, Theor. Chem. Acc. 115, 330 (2006): $E_{\rm HF,n}=E_{\rm HF,CBS} +A (n+1)\cdot \exp(-9 \sqrt{n})$. Use METHOD_R=KM for this.

The following example shows various possibilities for extrapolation:

examples/h2o_extrapolate_ccsd.inp
***,h2o

gthresh,energy=1.d-9
basis=avtz

r =   0.9572 ang, theta =  104.52
geometry={!nosym
O;
H1,O,r;
H2,O,r,H1,theta}

hf
{ccsd(t)}
text,compute energies, extrapolate reference energy using EX1 and correlation energy using L3
extrapolate,basis=avtz:avqz:av5z,method_c=l3,method_r=ex1,npc=2

ehf=energr(1:3)
etot=energy(1:3)

text,extrapolate total energy using EX2
extrapolate,basis=avtz:avqz:av5z,etot=etot,method=ex2

text,extrapolate reference energy by EX1 and correlation energy by EX2
extrapolate,basis=avtz:avqz:av5z,etot=etot,method_c=ex2,eref=ehf,method_r=ex1

text,extrapolate reference energy by EX1 and correlation energy by LH3
extrapolate,basis=avtz:avqz:av5z,etot=etot,method_c=LH3,eref=ehf,method_r=ex1,npc=2

text,extrapolate reference energy by EX1 and correlation energy by LX
extrapolate,basis=avtz:avqz:av5z,etot=etot,method_c=LX,eref=ehf,method_r=ex1,xc=[3,4],pc=0.5

The second example shows extrapolations of MRCI energies. In this case both the MRCI and the MRCI+Q energies are extrapolated.

examples/h2o_extrapolate_mrci.inp
***,h2o

gthresh,energy=1.d-9
basis=avtz

r =   0.9572 ang, theta =  104.52
geometry={
O;
H1,O,r;
H2,O,r,H1,theta}

hf
multi
mrci
text,Compute energies, extrapolate reference energy using EX1 and correlation energy using L3;
text,The Davidson corrected energy is also extraplated
extrapolate,basis=avtz:avqz:av5z,method_c=l3,method_r=ex1,npc=2

emc=energr
ecorr_mrci=energy-emc
ecorr_mrciq=energd-emc

text,Extrapolate reference energy by EX1 and correlation energy by LH3
text,The Davidson corrected energy is also extraplated
extrapolate,basis=avtz:avqz:av5z,ecorr=ecorr_mrci,ecorrd=ecorr_mrciq,method_c=LH3,eref=emc,method_r=ex1,npc=2

Geometry optimizations are possible by using numerical gradients obtained from extrapolated energies. Analytical energy gradients cannot be used.

The following possibilities exist:

1.) If OPTG directly follows the EXTRAPOLATE command, the extrapolated energy is optimized automatically (only variable settings may occur between EXTRAPOLATE and OPTG).

Examples:

Extrapolating the energy for the last command:

examples/h2o_extrapol_opt1.inp
geometry={o;h1,o,r;h2,o,r,h1,theta}
theta=102
r=0.96 ang
basis=vtz

hf
ccsd(t)
extrapolate,basis=vtz:vqz

optg

Extrapolating the energy computed in a procedure:

examples/h2o_extrapol_opt2.inp
geometry={o;h1,o,r;h2,o,r,h1,theta}
theta=102
r=0.96 ang

proc ccsdt
hf
ccsd(t)
endproc

extrapolate,basis=vtz:vqz,proc=ccsdt

optg

Note that this is not possible if EXTRAPOLATE gets the input energies from variables.

2.) Using a procedure for the extrapolation:

By default, variable ECBS is optimized, but other variables (e.g. ECBSD) can be specified using the VARIABLE option on the OPTG command.

examples/h2o_extrapol_opt3.inp
geometry={o;h1,o,r;h2,o,r,h1,theta}
theta=102
r=0.96 ang
basis=vtz

proc cbs34
hf
ccsd(t)
extrapolate,basis=vtz:vqz
endproc

optg,variable=ecbs,proc=cbs34
examples/h2o_extrapol_opt4.inp
geometry={o;h1,o,r;h2,o,r,h1,theta}
theta=102
r=0.96 ang

proc cbs34
basis=vtz
hf
ccsd(t)
eref(1)=energr
ecc(1)=energy

basis=vqz
hf
ccsd(t)
eref(2)=energr
ecc(2)=energy
extrapolate,basis=vtz:vqz,eref=eref,etot=ecc
endproc

optg,variable=ecbs,proc=cbs34

This is possible by defining the extrapolation in a procedure:

examples/h2o_extrapol_freq.inp
geometry={o;h1,o,r;h2,o,r,h1,theta}
theta=102
r=0.96 ang
basis=vtz

proc cbs34
hf
ccsd(t)
extrapolate,basis=vtz:vqz
endproc

optg,variable=ecbs,proc=cbs34
freq,variable=ecbs,proc=cbs34

As an alternative to basis set extrapolations or using F12 correlation methods the BSIE (basis set incompleteness error) to electron correlation energies can also be corrected by using a density based model by Giner et al., see:

$[1]$ P.-F. Loos, B. Pradines, A. Scemama, J. Toulouse and E. Giner, A density-based basis-set correction for wave function theory, J. Phys. Chem. Lett. 10, 2931 (2019).
$[2]$ E. Giner, A. Scemama, P.-F. Loos and J. Toulouse, A basis-set error correction based on density-functional theory for strongly correlated molecular systems, J. Chem. Phys. 152, 174104 (2020).

This basis-set correction relies on a mapping between wave-function calculations in a finite basis set and range-separated DFT (RSDFT) through the definition of an effective non-divergent interaction corresponding to the electron–electron Coulomb interaction projected in the finite basis set. This enables the use of RSDFT-type complementary density functionals to recover the dominant part of the short-range correlation effects missing in this finite basis set.

To compute the extrapolated MP2 energy with the DFT correction the following input can be used for Molpro

!water molecule
r=0.9575 ANG
theta=104.51
geometry={O; H1,O,R; H2,O,R,H1,theta}

!small basis set with diffuse functions
basis=avdz

!hf calculation
hf
eref=energy !save reference energy

!mp2 calculation for this small basis set
mp2
ecorr=energy-eref !save mp2 correlation energy

!compute DFT correction
bascorr,core=1   !use option for frozen-core here

!add correction to total energy (DFT correction stored in ecmd variable
etot=eref+ecorr+ecmd

Note that the DFT correction may be used in conjunction with any wave function correlation method which suffers from the wrong description of the interelectronic cusp with finite basis sets, so in the above example the MP2 method could be replaced with any other electron correlation method available in Molpro.

The reference energy term is not corrected for the BSIE in the example above. One may do so by using either the basis set extrapolation approach described in this section:

hf
extrapolate,basis=avdz:avtz:avqz
eref=energr(4) !the last element in the array (first 3 elements are the avdz/avtz/avqz energies)

or by adding the CABS singles correction to the HF energy computed in the small basis set

hf
eref0=energy
df-mp2-f12,cabs_singles=-1
eref=eref0+ef12_singles

In general the density based BSIE correction can be called as

bascorr,<core=...>,<orb=...>,<tolorb=...>

with the following options

• CORE number of core orbitals (normally required, since by default all orbitals are included). When symmetry is used an array of the form [nc1,nc2,…] must be given to the option for specifying the core orbitals in each symmetry
• ORB the record storing the orbital coefficients. If omitted, the orbitals from the last SCF calculation are being used.
• TOLORB screening threshold for the calculation of the basis functions on the grid.

Since the calculation of the BSIE correction requires a numerical integration on a quadrature grid, the grid setting described here also can affect the results of the correction. In practice, however, the default settings for DFT integrations should be adequate.

The density absed BSIE correction involves the calculation of a 4-indexed quantity on the quadrature grid. For larger molecules this step can therefore become very expensive and it is strongly advised to use the corresponding density fitting implementation of the method which can drastically reduce the computation time. The DF implemention currently can not generate the fitting basis sets automatically, so an input of the following kind needs to be used then:

basis={
set,orbital; default,avdz
set,jkfit; default,avdz/jkfit   !only for scf calculation
set,mp2fit; default,avdz/mp2fit
}

df-hf,basis=jkfit
{cfit,basis=mp2fit}  !change to mp2fit basis set
df-bascorr

where the MP2Fit fitting basis set is used for the BASCORR program which has been found to yield better results than the correponding JKFit basis set that we use for the SCF calculation in the example above.