The many dimensions of electronic wave functions makes them difficult to analyze and interpret. It is often convenient to reduce this large number of dimensions, yielding simpler functions that can more readily provide chemical insight. The most familiar of these is the oneelectron density , which gives the probability of an electron being found at the point . Analogously, the oneelectron momentum density gives the probability that an electron will have a momentum of . However, the wave function is reduced to the oneelectron density much information is lost. In particular, it is often desirable to retain explicit twoelectron information. Intracules are twoelectron distribution functions and provide information about the relative position and momentum of electrons. A detailed account of the different type of intracules can be found in Ref. Gill:2003. QChem’s intracule package was developed by Aaron Lee and Nick Besley, and can compute the following intracules for or HF wave functions:
Position intracules, : describes the probability of finding two electrons separated by a distance .
Momentum intracules, : describes the probability of finding two electrons with relative momentum .
Wigner intracule, : describes the combined probability of finding two electrons separated by and with relative momentum .
The intracule density, , represents the probability for the interelectronic vector :
(11.17) 
where is the twoelectron density. A simpler quantity is the spherically averaged intracule density,
(11.18) 
where is the angular part of , measures the probability that two electrons are separated by a scalar distance . This intracule is called a position intracule.[Gill et al.(2003)Gill, O’Neill, and Besley] If the molecular orbitals are expanded within a basis set
(11.19) 
The quantity can be expressed as
(11.20) 
where is the twoparticle density matrix and is the position integral
(11.21) 
and , , and are basis functions. For HF wave functions, the position intracule can be decomposed into a Coulomb component,
(11.22) 
and an exchange component,
(11.23) 
where etc. are density matrix elements. The evaluation of , and within QChem has been described in detail in Ref. Lee:1999.
Some of the moments of are physically significant,[Gill(1997)] for example
(11.24)  
(11.25)  
(11.26)  
(11.27) 
where is the number of electrons and, is the electronic dipole moment and is the trace of the electronic quadrupole moment tensor. QChem can compute both moments and derivatives of position intracules.
Analogous quantities can be defined in momentum space; , for example, represents the probability density for the relative momentum :
(11.28) 
where momentum twoelectron density. Similarly, the spherically averaged intracule
(11.29) 
where is the angular part of , is a measure of relative momentum and is called the momentum intracule. The quantity can be written as
(11.30) 
where is the twoparticle density matrix and is the momentum integral[Besley et al.(2002)Besley, Lee, and Gill]
(11.31) 
The momentum integrals only possess fourfold permutational symmetry, i.e.,
(11.32)  
(11.33) 
and therefore generation of is roughly twice as expensive as . Momentum intracules can also be decomposed into Coulomb and exchange components:
(11.34) 
(11.35) 
Again, the evenorder moments are physically significant:[Besley et al.(2002)Besley, Lee, and Gill]
(11.36) 
(11.37) 
(11.38) 
(11.39) 
where is the number of electrons and is the total electronic kinetic energy. Currently, QChem can compute , and using and basis functions only. Moments are generated using quadrature and consequently for accurate results must be computed over a large and closely spaced range.
The intracules and provide a representation of an electron distribution in either position or momentum space but neither alone can provide a complete description. For a combined position and momentum description an intracule in phase space is required. Defining such an intracule is more difficult since there is no phase space secondorder reduced density. However, the secondorder Wigner distribution,[Besley et al.(2003)Besley, O’Neill, and Gill]
(11.40) 
can be interpreted as the probability of finding an electron at with momentum and another electron at with momentum . [The quantity is often referred to as “quasiprobability distribution” since it is not positive everywhere.]
The Wigner distribution can be used in an analogous way to the second order reduced densities to define a combined position and momentum intracule. This intracule is called a Wigner intracule, and is formally defined as
(11.41) 
If the orbitals are expanded in a basis set, then can be written as
(11.42) 
where ( is the Wigner integral
(11.43) 
Wigner integrals are similar to momentum integrals and only have fourfold permutational symmetry. Evaluating Wigner integrals is considerably more difficult that their position or momentum counterparts. The fundamental integral,
(11.44)  
(11.45) 
can be expressed as
(11.46) 
or alternatively
(11.47) 
Two approaches for evaluating have been implemented in QChem, full details can be found in Ref. Wigner:1932. The first approach uses the first form of and used Lebedev quadrature to perform the remaining integrations over . For high accuracy large Lebedev grids[Lebedev(1976), Lebedev(1977), Lebedev and Laikov(1999)] should be used, grids of up to 5294 points are available in QChem. Alternatively, the second form can be adopted and the integrals evaluated by summation of a series. Currently, both methods have been implemented within QChem for and basis functions only.
When computing intracules it is most efficient to locate the loop over and/or points within the loop over shellquartets.[Cioslowski and Liu(1996)] However, for this requires a large amount of memory to store all the integrals arising from each point. Consequently, an additional scheme, in which the and points loop is outside the shellquartet loop, is available. This scheme is less efficient, but substantially reduces the memory requirements.
The following $rem variables can be used to control the calculation of intracules.
INTRACULE
Controls whether intracule properties are calculated (see also the $intracule section).
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE
No intracule properties.
TRUE
Evaluate intracule properties.
RECOMMENDATION:
None
WIG_MEM
Reduce memory required in the evaluation of .
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE
Do not use low memory option.
TRUE
Use low memory option.
RECOMMENDATION:
The low memory option is slower, so use the default unless memory is limited.
WIG_LEB
Use Lebedev quadrature to evaluate Wigner integrals.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE
Evaluate Wigner integrals through series summation.
TRUE
Use quadrature for Wigner integrals.
RECOMMENDATION:
None
WIG_GRID
Specify angular Lebedev grid for Wigner intracule calculations.
TYPE:
INTEGER
DEFAULT:
194
OPTIONS:
Lebedev grids up to 5810 points.
RECOMMENDATION:
Larger grids if high accuracy required.
N_WIG_SERIES
Sets summation limit for Wigner integrals.
TYPE:
INTEGER
DEFAULT:
10
OPTIONS:
RECOMMENDATION:
Increase for greater accuracy.
N_I_SERIES
Sets summation limit for series expansion evaluation of .
TYPE:
INTEGER
DEFAULT:
40
OPTIONS:
RECOMMENDATION:
Lower values speed up the calculation, but may affect accuracy.
N_J_SERIES
Sets summation limit for series expansion evaluation of .
TYPE:
INTEGER
DEFAULT:
40
OPTIONS:
RECOMMENDATION:
Lower values speed up the calculation, but may affect accuracy.
int_type 
0 
Compute only 
1 
Compute only 

2 
Compute only 

3 
Compute , and 

4 
Compute and 

5 
Compute and 

6 
Compute and 

u_points 
Number of points, start, end. 

v_points 
Number of points, start, end. 

moments 
0–4 
Order of moments to be computed ( only). 
derivs 
0–4 
order of derivatives to be computed ( only). 
accuracy 

( specify accuracy of intracule interpolation table ( only). 
Example 11.252 Compute HF/STO3G , and for Ne, using Lebedev quadrature with 974 point grid.
$molecule 0 1 Ne $end $rem METHOD hf BASIS sto3g INTRACULE true WIG_LEB true WIG_GRID 974 $end $intracule int_type 3 u_points 10 0.0 10.0 v_points 8 0.0 8.0 moments 4 derivs 4 accuracy 8 $end
Example 11.253 Compute HF/631G intracules for HO using series summation up to =25 and 30 terms in the series evaluations of and .
$molecule 0 1 H1 O H1 r H2 O r H1 theta r = 1.1 theta = 106 $end $rem METHOD hf BASIS 631G INTRACULE true WIG_MEM true N_WIG_SERIES 25 N_I_SERIES 40 N_J_SERIES 50 $end $intracule int_type 2 u_points 30 0.0 15.0 v_points 20 0.0 10.0 $end