Excited States from Time-Dependent Density Functional Theory - Reviews in Computational Chemistry - page 103

In a finite basis set, all operators become finite matrices; the matrix
elements are integrals, as illustrated, for example, by Eq. [21]:

ð d 3 r

H

mns ½

n ¼ w m ð r Þ H

s ½

n w n ð r Þ ½ 21

The calculation and processing of such integrals is the main effort in almost
allDFTnumericalcalculations.Gaussian basis functions have the distinct
advantage that (1) most integrals can be evaluated analytically, and (2)
they are spatially local. The latter advantage implies that many integrals
vanish and need not be calculated. Whether a certain integral vanishes or
not can be decided in advance by prescreening techniques. 178
To illustrate the effect of basis set selection, we show in Table 2 the
reaction energy for naphthalene combustion in the gas phase:

C 10 H 8 þ 12O 2 ! 10CO 2 þ 4H 2 O ½ 22

The basis sets are listed in order of increasing size and are well known in
quantum chemistry (and are described in detail later in the chapter). We see that
hydrogen polarization functions (basis sets ending in P) are important because
C-H bonds are broken and O-H bonds are formed. Augmentation (aug-) with
diffuse functions (small exponent) improves somewhat the smaller basis set
results but is not economical in this case. Using the resolution of the identity
(RI) for the Coulomb operator saves computational time, with no loss of

Table 2 Single-Point Calculations (with the PBE/TZVP/RI Geometry) Using PBE
Functional for Reaction Energy for Naphthalene Combustion
Basis Set CPU a Negative Reaction Energy (kcal/mol) (s)

SV 916.8 9

SV(P) 1060.0 9

6-31G* 1047.1 25

SVP 1108.6 9

aug-SV(P) 1115.5 35

TZVP 1124.5 295

TZVPP 1131.2 420

cc-pVTZ 1129.0 375

aug-TZVP 1130.2 2525

aug-TZVP/RI 1130.2 155

QZVP 1140.3 5734

Reference Value b 1216.3

a CPU denotes the CPU time is seconds.
b Computed using standard enthalpies of formation (from NIST 179 ) using thermal and ZVPE
corrections at the PBE/TZVP/RI level.