# STO-nG basis sets

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 $mathbf phi_1 = left \left(frac\left\{2alpha_1\right\}\left\{pi\right\} ight \right) ^\left\{0.75\right\}e^\left\{-alpha_1 r^2\right\}$. 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

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$mathbf hat\left\{H\right\}_e = -frac\left\{ abla^2\right\}\left\{2\right\}-frac\left\{Z\right\}\left\{r\right\}$.
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,

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

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

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

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,*)" Basis set : " WRITE(2,002)"ALPHA(i)","C(i)" DO i=1,n WRITE(2,003)alpha(i),c(i) ENDDO WRITE(2,001)" K.E. integral is :", TI," hartree" WRITE(2,001)" P.E. integral is :", VI," hartree" WRITE(2,001)" Overlap Integral is :", SI," hartree" WRITE(2,001)" Energy of H atom is :", (VI+TI)/SI," hartree/partic &le" WRITE(2,001)" ENERGY of H atom is :",(VI+TI)*27.211397D0/SI," e.V &./particle" WRITE(2,001)" ENERGY of H atom is :",(VI+TI)*627.509D0/SI," kcal/ &mol" WRITE(2,001)" ENERGY of H atom is :",(VI+TI)*2625.51D0/SI," kJ/mo &l" WRITE(2,001)" ENERGY 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

*Quantum chemistry computer programs

References :

[1] http://gnode2.pnl.gov/bse/portal

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