A (hopefully) gentle guide to the computer implementation of molecular integrals over Gaussian basis functions.

Note: You can find the following integral routines implemented in a working Hartree-Fock code here.

PDF version

In quantum chemistry, we are often interested in evaluating integrals over Gaussian basis functions. Here I am going to take much of the theory for granted and talk about how one might actually implement the integrals for quantum chemistry by using recursion relationships with Hermite Gaussians. Our goal is to have clean, readable code. I’m writing this for the people are comfortable with the mathematics behind the Gaussian integrals, but want to see a readable computer implementation. I’ll talk a bit about some computational considerations at the end, but my goal is to convert equations to code. In fact, I’ve tried to structure the equations and the code in such a way that the two look very similar.

For a very good overview of integral evaluation, please see:

Helgaker, Trygve, and Peter R. Taylor. “Gaussian basis sets and molecular integrals.” Modern Electronic Structure (1995).

I will try and follow the notation used in the above reference.

Mathematical preliminaries

Let’s start with some of the basics. First, we have our 3D Gaussian functions

with orbital exponent , electronic coordinates , origin , and

also, are the angular quantum numbers (e.g. is -type, is type, etc.) Cartesian Gaussians are separable in 3D along so that

with the 1D Gaussian

So far, so good. Let’s consider the overlap integral of two 1D Gaussians, and

where we used the Gaussian product theorem so that

When using Hermite Gaussians, we can express as

where are expansion coefficients (to be determined recursively) and is the Hermite Gaussian overlap of two Gaussians and . It has a simple expression that kills the sum via the Kronecker delta . It can be shown that the expansion coefficients can be defined using the following recursive definitions

The first equation gives us a way to reduce the index and the second gives us a way to reduce index so that we can get to the third equation, which is our base case. The last equation tells us what to do if we go out of index bounds.

Overlap integrals

The first thing we need to do is implement a function E which returns our expansion coefficients . Aside from angular momentum and from the Gaussian functions, we also need the distance between Gaussians and the orbital exponent coefficients and as inputs.

def E(i,j,t,Qx,a,b):
    ''' Recursive definition of Hermite Gaussian coefficients.
        Returns a float.
        a: orbital exponent on Gaussian 'a' (e.g. alpha in the text)
        b: orbital exponent on Gaussian 'b' (e.g. beta in the text)
        i,j: orbital angular momentum number on Gaussian 'a' and 'b'
        t: number nodes in Hermite (depends on type of integral, 
           e.g. always zero for overlap integrals)
        Qx: distance between origins of Gaussian 'a' and 'b'
    '''
    p = a + b
    q = a*b/p
    if (t < 0) or (t > (i + j)):
        # out of bounds for t  
        return 0.0
    elif i == j == t == 0:
        # base case
        return np.exp(-q*Qx*Qx) # K_AB
    elif j == 0:
        # decrement index i
        return (1/(2*p))*E(i-1,j,t-1,Qx,a,b) - \
               (q*Qx/a)*E(i-1,j,t,Qx,a,b)    + \
               (t+1)*E(i-1,j,t+1,Qx,a,b)
    else:
        # decrement index j
        return (1/(2*p))*E(i,j-1,t-1,Qx,a,b) + \
               (q*Qx/b)*E(i,j-1,t,Qx,a,b)    + \
               (t+1)*E(i,j-1,t+1,Qx,a,b)

This is simple enough! So for a 1D overlap between two Gaussians we would just need to evaluate and multiply it by . Overlap integrals in 3D are just a product of the 1D overlaps. We could imagine a 3D overlap function like so

import numpy as np

def overlap(a,lmn1,A,b,lmn2,B):
    ''' Evaluates overlap integral between two Gaussians
        Returns a float.
        a:    orbital exponent on Gaussian 'a' (e.g. alpha in the text)
        b:    orbital exponent on Gaussian 'b' (e.g. beta in the text)
        lmn1: int tuple containing orbital angular momentum (e.g. (1,0,0))
              for Gaussian 'a'
        lmn2: int tuple containing orbital angular momentum for Gaussian 'b'
        A:    list containing origin of Gaussian 'a', e.g. [1.0, 2.0, 0.0]
        B:    list containing origin of Gaussian 'b'
    '''
    l1,m1,n1 = lmn1 # shell angular momentum on Gaussian 'a'
    l2,m2,n2 = lmn2 # shell angular momentum on Gaussian 'b'
    S1 = E(l1,l2,0,A[0]-B[0],a,b,n[0],A[0]) # X
    S2 = E(m1,m2,0,A[1]-B[1],a,b,n[1],A[1]) # Y
    S3 = E(n1,n2,0,A[2]-B[2],a,b,n[2],A[2]) # Z
    return S1*S2*S3*np.power(np.pi/(a+b),1.5) 

Note that we are using the NumPy package in order to take advantage of the definitions of and the fractional power to the . The above two functions overlap and E are enough to get us the overlap between two Gaussian functions (primitives), but most basis functions are contracted, meaning they are the sum of multiple Gaussian primitives. It is not too difficult to account for this, and we can finally wrap up our evaluation of overlap integrals with a function S(a,b) which returns the overlap integral between two contracted Gaussian functions.

def S(a,b):
    '''Evaluates overlap between two contracted Gaussians
       Returns float.
       Arguments:
       a: contracted Gaussian 'a', BasisFunction object
       b: contracted Gaussian 'b', BasisFunction object
    '''
    s = 0.0
    for ia, ca in enumerate(a.coefs):
        for ib, cb in enumerate(b.coefs):
            s += a.norm[ia]*b.norm[ib]*ca*cb*\
                     overlap(a.exps[ia],a.shell,a.origin,
                     b.exps[ib],b.shell,b.origin)
    return s

Basically, this is just a sum over primitive overlaps, weighted by normalization and coefficient. A word is in order for the arguments, however. In order to keep the number of arguments we have to pass into our functions, we have created BasisFunction objects that contain all the relevant data for the basis function, including exponents, normalization, etc. A BasisFunction class looks like

class BasisFunction(object):
    ''' A class that contains all our basis function data
        Attributes:
        origin: array/list containing the coordinates of the Gaussian origin
        shell:  tuple of angular momentum
        exps:   list of primitive Gaussian exponents
        coefs:  list of primitive Gaussian coefficients
        norm:   list of normalization factors for Gaussian primitives
    '''
    def __init__(self,origin=[0.0,0.0,0.0],shell=(0,0,0),exps=[],coefs=[]):
        self.origin = np.asarray(origin)
        self.shell = shell
        self.exps  = exps
        self.coefs = coefs
        self.norm = None
        self.normalize()

    def normalize(self):
        ''' Routine to normalize the basis functions, in case they
            do not integrate to unity.
        '''
        l,m,n = self.shell
        # self.norm is a list of length equal to number primitives
        self.norm = np.sqrt(np.power(2,2*(l+m+n)+1.5)*
                        np.power(self.exps,l+m+n+1.5)/
                        fact2(2*l-1)/fact2(2*m-1)/
                        fact2(2*n-1)/np.power(np.pi,1.5))

So, for example if we had a STO-3G Hydrogen 1s at origin (1.0, 2.0, 3.0), we could create a basis function object for it like so

myOrigin = [1.0, 2.0, 3.0]
myShell  = (0,0,0) # p-orbitals would be (1,0,0) or (0,1,0) or (0,0,1), etc.
myExps   = [3.42525091, 0.62391373, 0.16885540] 
myCoefs  = [0.15432897, 0.53532814, 0.44463454]
a = BasisFunction(origin=myOrigin,shell=myShell,exps=myExps,coefs=myCoefs)

Where we used the EMSL STO-3G definition

!  STO-3G  EMSL  Basis Set Exchange Library 
! Elements                             References
! --------                             ----------
!  H - Ne: W.J. Hehre, R.F. Stewart and J.A. Pople, J. Chem. Phys. 2657 (1969).

****
H     0 
S   3   1.00
      3.42525091             0.15432897       
      0.62391373             0.53532814       
      0.16885540             0.44463454       
****

So doing S(a,a) = 1.0, since the overlap of a basis function with itself (appropriately normalized) is one.

Kinetic energy integrals

Having finished the overlap integrals, we move on to the kinetic integrals. The kinetic energy integrals can be written in terms of overlap integrals

where

For a 3D primitive, we can form a kinetic function analogous to overlap,

def kinetic(a,lmn1,A,b,lmn2,B):
    ''' Evaluates kinetic energy integral between two Gaussians
        Returns a float.
        a:    orbital exponent on Gaussian 'a' (e.g. alpha in the text)
        b:    orbital exponent on Gaussian 'b' (e.g. beta in the text)
        lmn1: int tuple containing orbital angular momentum (e.g. (1,0,0))
              for Gaussian 'a'
        lmn2: int tuple containing orbital angular momentum for Gaussian 'b'
        A:    list containing origin of Gaussian 'a', e.g. [1.0, 2.0, 0.0]
        B:    list containing origin of Gaussian 'b'
    '''
    l1,m1,n1 = lmn1
    l2,m2,n2 = lmn2
    term0 = b*(2*(l2+m2+n2)+3)*\
                            overlap(a,(l1,m1,n1),A,b,(l2,m2,n2),B)
    term1 = -2*np.power(b,2)*\
                           (overlap(a,(l1,m1,n1),A,b,(l2+2,m2,n2),B) +
                            overlap(a,(l1,m1,n1),A,b,(l2,m2+2,n2),B) +
                            overlap(a,(l1,m1,n1),A,b,(l2,m2,n2+2),B))
    term2 = -0.5*(l2*(l2-1)*overlap(a,(l1,m1,n1),A,b,(l2-2,m2,n2),B) +
                  m2*(m2-1)*overlap(a,(l1,m1,n1),A,b,(l2,m2-2,n2),B) +
                  n2*(n2-1)*overlap(a,(l1,m1,n1),A,b,(l2,m2,n2-2),B))
    return term0+term1+term2

and for contracted Gaussians we make our final function T(a,b)

def T(a,b):
    '''Evaluates kinetic energy between two contracted Gaussians
       Returns float.
       Arguments:
       a: contracted Gaussian 'a', BasisFunction object
       b: contracted Gaussian 'b', BasisFunction object
    '''
    t = 0.0
    for ia, ca in enumerate(a.coefs):
        for ib, cb in enumerate(b.coefs):
            t += a.norm[ia]*b.norm[ib]*ca*cb*\
                     kinetic(a.exps[ia],a.shell,a.origin,\
                     b.exps[ib],b.shell,b.origin)
    return t

Nuclear attraction integrals

The last one-body integral I want to consider here is the nuclear attraction integrals. These differ from the overlap and kinetic energy integrals in that the nuclear attraction operator is Coulombic, meaning we cannot easily factor the integral into Cartesian components .

To evaluate these integrals, we need to set up an auxiliary Hermite Coulomb integral that handles the Coulomb interaction between a Gaussian charge distribution centered at and a nuclei centered at . The Hermite Coulomb integral, like its counterpart , is defined recursively:

where is the Boys function

which is a special case of the Kummer confluent hypergeometric function,

which is convenient for us, since SciPy has an implementation of as a part of scipy.special. So for R we can code up the recursion like so

def R(t,u,v,n,p,PCx,PCy,PCz,RPC):
    ''' Returns the Coulomb auxiliary Hermite integrals 
        Returns a float.
        Arguments:
        t,u,v:   order of Coulomb Hermite derivative in x,y,z
                 (see defs in Helgaker and Taylor)
        n:       order of Boys function 
        PCx,y,z: Cartesian vector distance between Gaussian 
                 composite center P and nuclear center C
        RPC:     Distance between P and C
    '''
    T = p*RPC*RPC
    val = 0.0
    if t == u == v == 0:
        val += np.power(-2*p,n)*boys(n,T)
    elif t == u == 0:
        if v > 1:
            val += (v-1)*R(t,u,v-2,n+1,p,PCx,PCy,PCz,RPC)
        val += PCz*R(t,u,v-1,n+1,p,PCx,PCy,PCz,RPC)
    elif t == 0:
        if u > 1:
            val += (u-1)*R(t,u-2,v,n+1,p,PCx,PCy,PCz,RPC)
        val += PCy*R(t,u-1,v,n+1,p,PCx,PCy,PCz,RPC)
    else:
        if t > 1:
            val += (t-1)*R(t-2,u,v,n+1,p,PCx,PCy,PCz,RPC)
        val += PCx*R(t-1,u,v,n+1,p,PCx,PCy,PCz,RPC)
    return val

and we can define our boys(n,T) function as

from scipy.special import hyp1f1

def boys(n,T):
    return hyp1f1(n+0.5,n+1.5,-T)/(2.0*n+1.0) 

There are other definitions of the Boys function of course, in case you do not want to use the SciPy built-in. Note that R requires knowledge of the composite center from two Gaussians centered at and . We can determine using the Gaussian product center rule

which is very simply coded up as

def gaussian_product_center(a,A,b,B):
    return (a*A+b*B)/(a+b)

Now that we have a the Coulomb auxiliary Hermite integrals , we can form the nuclear attraction integrals with respect to a given nucleus centered at , , via the expression

def nuclear_attraction(a,lmn1,A,b,lmn2,B,C):
    ''' Evaluates kinetic energy integral between two Gaussians
         Returns a float.
         a:    orbital exponent on Gaussian 'a' (e.g. alpha in the text)
         b:    orbital exponent on Gaussian 'b' (e.g. beta in the text)
         lmn1: int tuple containing orbital angular momentum (e.g. (1,0,0))
               for Gaussian 'a'
         lmn2: int tuple containing orbital angular momentum for Gaussian 'b'
         A:    list containing origin of Gaussian 'a', e.g. [1.0, 2.0, 0.0]
         B:    list containing origin of Gaussian 'b'
         C:    list containing origin of nuclear center 'C'
     '''
     l1,m1,n1 = lmn1 
     l2,m2,n2 = lmn2
     p = a + b
     P = gaussian_product_center(a,A,b,B) # Gaussian composite center
     RPC = np.linalg.norm(P-C)
 
     val = 0.0
     for t in xrange(l1+l2+1):
         for u in xrange(m1+m2+1):
             for v in xrange(n1+n2+1):
                 val += E(l1,l2,t,A[0]-B[0],a,b) * \
                        E(m1,m2,u,A[1]-B[1],a,b) * \
                        E(n1,n2,v,A[2]-B[2],a,b) * \
                        R(t,u,v,0,p,P[0]-C[0],P[1]-C[1],P[2]-C[2],RPC)
     val *= 2*np.pi/p 
     return val

And, just like all the other routines, we can wrap it up to treat contracted Gaussians like so:

def V(a,b,C):
    '''Evaluates overlap between two contracted Gaussians
        Returns float.
        Arguments:
        a: contracted Gaussian 'a', BasisFunction object
        b: contracted Gaussian 'b', BasisFunction object
        C: center of nucleus
     '''
     v = 0.0
     for ia, ca in enumerate(a.coefs):
         for ib, cb in enumerate(b.coefs):
             v += a.norm[ia]*b.norm[ib]*ca*cb*\
                      nuclear_attraction(a.exps[ia],a.shell,a.origin,
                      b.exps[ib],b.shell,b.origin,C)
     return v

Important: Note that this is the nuclear repulsion integral contribution from an atom centered at . To get the full nuclear attraction contribution, you must sum over all the nuclei, as well as scale each term by the appropriate nuclear charge!

Two electron repulsion integrals

We are done with the necessary one-body integrals (for a basic Hartree-Fock energy code, at least) and are ready to move on to the two-body terms: the electron-electron repulsion integrals. Thankfully, much of the work has been done for us on account of the nuclear attraction one-body integrals.

In terms of Hermite integrals, to evaluate the two electron repulsion terms, we must evaluate the summation

which looks terrible and it is. However, recalling that letting (that is, the Gaussian exponents on and , and and ), we can write the equation in a similar form to the nuclear attraction integrals

def electron_repulsion(a,lmn1,A,b,lmn2,B,c,lmn3,C,d,lmn4,D):
    ''' Evaluates kinetic energy integral between two Gaussians
         Returns a float.
         a,b,c,d:   orbital exponent on Gaussian 'a','b','c','d'
         lmn1,lmn2
         lmn3,lmn4: int tuple containing orbital angular momentum
                    for Gaussian 'a','b','c','d', respectively
         A,B,C,D:   list containing origin of Gaussian 'a','b','c','d'
     '''
     l1,m1,n1 = lmn1
     l2,m2,n2 = lmn2
     l3,m3,n3 = lmn3
     l4,m4,n4 = lmn4
     p = a+b # composite exponent for P (from Gaussians 'a' and 'b')
     q = c+d # composite exponent for Q (from Gaussians 'c' and 'd')
     alpha = p*q/(p+q)
     P = gaussian_product_center(a,A,b,B) # A and B composite center
     Q = gaussian_product_center(c,C,d,D) # C and D composite center
     RPQ = np.linalg.norm(P-Q)
 
     val = 0.0
     for t in xrange(l1+l2+1):
         for u in xrange(m1+m2+1):
             for v in xrange(n1+n2+1):
                 for tau in xrange(l3+l4+1):
                     for nu in xrange(m3+m4+1):
                         for phi in xrange(n3+n4+1):
                             val += E(l1,l2,t,A[0]-B[0],a,b) * \
                                    E(m1,m2,u,A[1]-B[1],a,b) * \
                                    E(n1,n2,v,A[2]-B[2],a,b) * \
                                    E(l3,l4,tau,C[0]-D[0],c,d) * \
                                    E(m3,m4,nu ,C[1]-D[1],c,d) * \
                                    E(n3,n4,phi,C[2]-D[2],c,d) * \
                                    np.power(-1,tau+nu+phi) * \
                                    R(t+tau,u+nu,v+phi,0,\
                                        alpha,P[0]-Q[0],P[1]-Q[1],P[2]-Q[2],RPQ)
 
     val *= 2*np.power(np.pi,2.5)/(p*q*np.sqrt(p+q))
     return val

And, for completeness’ sake, we wrap the above to handle contracted Gaussians

def ERI(a,b,c,d):
    '''Evaluates overlap between two contracted Gaussians
        Returns float.
        Arguments:
        a: contracted Gaussian 'a', BasisFunction object
        b: contracted Gaussian 'b', BasisFunction object
        c: contracted Gaussian 'b', BasisFunction object
        d: contracted Gaussian 'b', BasisFunction object
     '''
     eri = 0.0
     for ja, ca in enumerate(a.coefs):
         for jb, cb in enumerate(b.coefs):
             for jc, cc in enumerate(c.coefs):
                 for jd, cd in enumerate(d.coefs):
                     eri += a.norm[ja]*b.norm[jb]*c.norm[jc]*d.norm[jd]*\
                              ca*cb*cc*cd*\
                              electron_repulsion(a.exps[ja],a.shell,a.origin,\
                                                 b.exps[jb],b.shell,b.origin,\
                                                 c.exps[jc],c.shell,c.origin,\
                                                 d.exps[jd],d.shell,d.origin)
     return eri

And there you have it! All the integrals necessary for a Hartree-Fock SCF code.

Computational efficiency considerations

Our goal here has been to eliminate some of the confusion when it comes to connecting mathematics to actual computer code. So the code that we have shown is hopefully clear and looks nearly identical to the mathematical equations they are supposed to represent. This is one reason we chose Python as the code of choice to implement the integrals. It emphasizes readability.

If you use the code as is, you’ll find that you can only really handle small systems. To that end, I’ll give a few ideas on how to improve the integral code to actually be usable.

First, I would recommend not using pure Python. The problem is that we have some pretty deep loops in the code, and nested for loops will kill your speed if you insist on sticking with Python. Now, you can code in another language, but I would suggest rewriting some of the lower level routines with Cython (http://www.cython.org). Cython statically compiles your code to eliminate many of the Python calls that slow your loops down. In my experience, you will get several orders of magnitude speed up. This brings me to my second point. One of the problems with Python is that you have the global interpreter lock (GIL) which basically means you cannot run things in parallel. Many of the integral evaluation routines would do well if you could split the work up over multiple CPUs. If you rewrite some of the low level routines such that they do not make Python calls anymore, you can turn off the GIL and wrap the code in OpenMP directives, or even use Cython’s cython.parallel module. This will take some thought, but can definitely be done. Furthermore, removing the explicit recursion in the E and R functions and making them iterative would go a long way to speed up the code.

A couple other thoughts: be sure to exploit the permutational symmetry of the integrals. The two electron repulsion integrals, for example, can be done in of the time just by exploiting these symmetries, which are unrelated to point group. Also, you can exploit many of the integral screening routines, since many of the two electron integrals are effectively zero. There are a lot of tricks out there in the literature, go check them out!