Definitions

STO-nG basis sets

STO-nG basis sets

STO-nG basis sets are the minimal basis sets, where 'n' represents the number of primitive Gaussian functions comprising a single basis set. For minimal basis sets, the core and valence orbitals are represented by same number primitive Gaussian functions $mathbf phi_i$. For example, an STO-3G basis set for the 1s orbital of H atom is a linear combination of 3 primitive Gaussian functions. It is easy to calculate the energy of an electron in the 1s orbital of H atom represented by STO-nG basis sets. In the following sections, the structure of the STO-nG minimal basis sets are explained with H atom as an example.

STO-1G basis set

$mathbf psi\left(1s_H\right)= psi_\left\{STO-1G\right\}=c_1phi_1$, where $mathbf c_1 = 1$ and $mathbf phi_1 = left \left(frac\left\{2alpha_1\right\}\left\{pi\right\} right \right) ^\left\{0.75\right\}e^\left\{-alpha_1 r^2\right\}$. The optimum value of $mathbf alpha_1$ is the one which gives the minimum value for the Energy of the 1s electron of H atom. The exponent $mathbf alpha_1$ for the STO-1G basis set can be manually derived by equating the derivative of the energy with respect to the exponent to zero.
Thus $mathbf alpha_1 = frac\left\{8 Z^2\right\}\left\{9 pi\right\} = 0.28294212$ and for the value $mathbf alpha_1 = 0.28294212$, the energy of the 1s electron of H atom can be calculated as $mathbf -0.42441318$ hartree. The expression for the energy of the 1s electron of H atom is a function only of $mathbf c_1$, $mathbf alpha_1$ and other fundamental constants such as $mathbf pi$. For convenience, the basis set details can be represented as follows
STO-1G $mathbf alpha$ $mathbf c$
0.2829421200D+00 1.0000000000D+00

STO-2G basis set

In general an STO-nG basis set is a linear combination of n primitive Gaussian functions. The STO-nG basis sets are usually represented by the exponents and the corresponding coefficients. Thus an STO-2G [Ref. 1] basis set which is a linear combination of 2 primitive Gaussian functions can be represented as follows.
STO-2G $mathbf alpha$ $mathbf c$
0.1309756377D+01 0.4301284983D+00
0.2331359749D+00 0.6789135305D+00

Accuracy

The exact energy of the 1s electron of H atom is -0.5 hartree. Following table illustrates the increase in accuracy as the number of primitive Gaussian functions increases in the basis set.
Basis set Energy [hartree]
STO-1G -0.424413182
STO-2G [Ref. 1] -0.454397402
STO-3G [Ref. 1] -0.466581850
STO-4G -0.469806464
STO-5G -0.470742918
STO-6G [Ref. 1] -0.471039054

Calculation of electronic energy using STO-nG basis sets (For ex. H atom)

The electronic energy of a molecular system is calculated as the expectation value of the molecular electronic Hamiltonian :

,

where $mathbf hat\left\{H\right\}_e$ is the electronic hamiltonian of the molecule. The expectation values can be analytically solved only for a two body system such as a Hydrogen atom. The electronic Hamiltonian for H-atom is given by $mathbf hat\left\{H\right\}_e = -frac\left\{nabla^2\right\}\left\{2\right\}-frac\left\{Z\right\}\left\{r\right\}$.
The exact integrals for the kinetic energy, potential energy expectation values and overlap integrals can be obtained as follows

,

,

.

Now the total energy expectation value can be divided into 3 parts, the kinetic energy expectation value, the potential energy expectation value and the overlap integrals.

$mathbf E_\left\{elec\right\} = frac\left\{+\right\}\left\{S\right\}$ where,

$mathbf= frac \left\{6sqrt\left\{2\right\}sum_\left\{i=1\right\}^n sum_\left\{j=1\right\}^n c_i c_j \left(alpha_ialpha_j\right)^\left\{7/4\right\}\right\}\left\{\left(alpha_i+alpha_j\right)^\left\{5/2\right\}\right\}$,

$mathbf= frac \left\{-4sqrt\left\{2\right\}Z sum_\left\{i=1\right\}^n sum_\left\{j=1\right\}^n c_i c_j \left(alpha_ialpha_j\right)^\left\{3/4\right\}\right\}\left\{sqrt\left\{pi\right\}\left(alpha_i+alpha_j\right)\right\}$,

$mathbf S = frac \left\{2sqrt\left\{2\right\} sum_\left\{i=1\right\}^n sum_\left\{j=1\right\}^n c_i c_j \left(alpha_ialpha_j\right)^\left\{3/4\right\}\right\}\left\{\left(alpha_i+alpha_j\right)^\left\{3/2\right\}\right\}$.

Thus when an STO-nG basis set with n Gaussian promitives is used, there are $n^2$ kinetic energy integrals, $n^2$ potential energy integrals and $n^2$ overlap integrals. Thus with $n$ primitive GTFs in the basis set, we need $3n^2$ integrals.

Appendix

The basis sets STO-nG [n=2,3&6] can be referred from the online basis set exchange [Ref. 1] and the energy of the 1s electron of H atom can easily be calculated by hand or by using a small program. Following is a Fortran77 program where the energy expression is explicitly stated and by giving the basis set as the input, the energy value is obtained as output.

`     !----------------------------------------------------------------`
`     ! PROGRAM sto_ng CALCULATES THE ENERGY OF 1s ELECTRON OF "H" ATOM`
`     ! OR OTHER HYDROGENIC ATOMIC SYSTEMS WITH MINIMAL BASIS SETS. THE`
`     ! PROGRAM CAN BE EASILY EXTENDED FOR LARGER BASIS SETS.`
`     !----------------------------------------------------------------`
`     PROGRAM sto_ng`
`     IMPLICIT NONE`
`     !----------------------------------------------------------------`
`     ! i AND j : DUMMY INDICES`
`     !       n : NUMBER OF PRIMITIVE GTOs`
`     !       Z : ATOMIC NUMBER`
`     !----------------------------------------------------------------`
`     INTEGER i, j, n, Z`
`     !----------------------------------------------------------------`
`     !  V(i,j) : i,j TH ELEMENT OF THE POTENTIAL ENERGY MATRIX`
`     !  T(i,j) : i,j TH ELEMENT OF THE KINETIC ENERGY MATRIX`
`     !  S(i,j) : i,j TH ELEMENT OF THE OVERLAP INTEGRAL MATRIX`
`     !      VI : TOTAL SUM OF ALL POTENTIAL ENERGY INTEGRALS`
`     !      TI : TOTAL SUM OF ALL KINETIC ENERGY INTEGRALS`
`     !      SI : TOTAL SUM ALL OF OVERLAP INTEGRALS`
`     !    c(i) : i TH COEFFICIENT`
`     !alpha(i) : i TH EXPONENT`
`     !----------------------------------------------------------------`
`     DOUBLE PRECISION V(100,100), T(100,100), S(100,100)`
`     DOUBLE PRECISION alpha(100), c(100), VI, TI, SI, PI`
`     PI=3.1415926535898D0`
`     OPEN(UNIT=1, FILE="input.txt")`
`     OPEN(UNIT=2, FILE="output.txt")`
`     READ(1,*)Z,n`
`     DO i=1,n`
`        READ(1,*)alpha(i),c(i)`
`     ENDDO`
`    !----------------------------------------------------------------`
`    ! CALCULATION OF OVERLAP INTEGRALS AND THEIR SUMMATION`
`    !----------------------------------------------------------------`
`     DO i=1,n`
`        DO j=1,n`
`          S(i,j)=c(i)*c(j)*2.0D0*SQRT(2.0D0)*(alpha(i)*alpha(j))**0.75D`
`    &0/(alpha(i)+alpha(j))**(1.5D0)`
`        ENDDO`
`     ENDDO`
`     SI=0.0D0`
`     DO i=1,n`
`        DO j=1,n`
`           SI=SI+S(i,j)`
`       ENDDO`
`     ENDDO`
`    !----------------------------------------------------------------`
`    ! CALCULATION OF KINETIC ENERGY INTEGRALS AND THEIR SUMMATION`
`    !----------------------------------------------------------------`
`     DO i=1,n`
`        DO j=1,n`
`        T(i,j)=c(i)*c(j)*6.0D0*SQRT(2.0D0)*(alpha(i)*alpha(j))**1.75D0/`
`    &(alpha(i)+alpha(j))**(2.5D0)`
`       ENDDO`
`     ENDDO`
`     TI=0.0D0`
`     DO i=1,n`
`        DO j=1,n`
`           TI=TI+T(i,j)`
`       ENDDO`
`     ENDDO`
`    !----------------------------------------------------------------`
`    ! CALCULATION OF POTENTIAL ENERGY INTEGRALS AND THEIR SUMMATION`
`    !----------------------------------------------------------------`
`     DO i=1,n`
`        DO j=1,n`
`          V(i,j)=-c(i)*c(j)*4.0D0*SQRT(2.0D0)*Z*(alpha(i)*alpha(j))**0.`
`    &75D0/(SQRT(PI)*(alpha(i)+alpha(j)))`
`        ENDDO`
`     ENDDO`
`     VI=0.0D0`
`     DO i=1,n`
`        DO j=1,n`
`           VI=VI+V(i,j)`
`       ENDDO`
`     ENDDO`
`     WRITE(2,*)"nnBasis set :n"`
`     WRITE(2,002)"ALPHA(i)","C(i)"`
`     DO i=1,n`
`        WRITE(2,003)alpha(i),c(i)`
`     ENDDO`
`     WRITE(2,001)"nnK.E. integral is    :", TI," hartree"`
`     WRITE(2,001)"nP.E. integral is    :", VI," hartree"`
`     WRITE(2,001)"nOverlap Integral is :", SI," hartree"`
`     WRITE(2,001)"nEnergy of H atom is :", (VI+TI)/SI," hartree/partic`
`    &le"`
`     WRITE(2,001)"nENERGY of H atom is :",(VI+TI)*27.211397D0/SI," e.V`
`    &./particle"`
`     WRITE(2,001)"nENERGY of H atom is :",(VI+TI)*627.509D0/SI," kcal/`
`    &mol"`
`     WRITE(2,001)"nENERGY of H atom is :",(VI+TI)*2625.51D0/SI," kJ/mo`
`    &l"`
`     WRITE(2,001)"nENERGY of H atom is :",(VI+TI)*219475D0/SI," cm-1"`
` 001 FORMAT(A,D20.10,A)`
` 002 FORMAT(8X,A,19X,A)`
` 003 FORMAT(D20.10,6X,D20.10)`
` 004 FORMAT(D20.10)`
`     CLOSE(1)`
`     CLOSE(2)`
`     STOP`
`     END`

`     INPUT FILE DETAILS`
`     FILE : input.txt`
`      1                                       ! ATOMIC NUMBER`
`      2                                       ! NO. OF PRIMITIVE GTOs`
`      0.1309756377D+01  0.4301284983D+00      ! BASIS SET  alpha c`
`      0.2331359749D+00  0.6789135305D+00`

`     OUTPUT FILE DETAILS`
`     FILE : output.txt`
`     Basis set :`
`         ALPHA(i)                   C(i)`
`     0.1309756377E+01          0.4301284983E+00`
`     0.2331359749E+00          0.6789135305E+00`
`     K.E. integral is    :    0.7348827001E+00 hartree`
`     P.E. integral is    :   -0.1189280102E+01 hartree`
`     Overlap Integral is :    0.1000000000E+01 hartree`
`     Energy of H atom is :   -0.4543974016E+00 hartree/particle`
`     ENERGY of H atom is :   -0.1236478809E+02 e.V./particle`
`     ENERGY of H atom is :   -0.2851384591E+03 kcal/mol`
`     ENERGY of H atom is :   -0.1193024922E+04 kJ/mol`
`     ENERGY of H atom is :   -0.9972886973E+05 cm-1`