Augmenting basis sets by extra functions

In many cases, one may want to add more basis function to a given basis set in order to make it more suitable for properties it was not designed to reproduce. Common examples are the improved description of excited states or of very weakly bound electrons by adding more diffuse functions into augmented basis sets, or the improved description of the wave function close to the nucleus by adding more steep functions into a basis set.

The extrapolation works by taking the two outermost primitives in the basis, $X$ and $Y$. We assume $X$ to be the steepest or most diffuse, and $Y$ is the second-steepest or second-most diffuse. (If you want to do steep augmentation, you should also fully decontract your basis set.) Now, we can just add more primitives as $X(X/Y)$, $X(X/Y{)}^2$, … The procedure is the best justified for even-tempered basis sets, where all exponents follow the same pattern, but the procedure is used often also for fully optimized basis sets.

Adding more diffuse functions to an already augmented basis set is generally known as multiple augmentation, as described by Woon & Dunning (https://doi.org/10.1063/1.466439). Adding a single function per angular momentum shell to an aug set leads to a doubly augmented (d-aug) set; adding two leads to a triply augmented (t-aug) set, and so on.

This functionality is implemented in the basis_set_exchange.manip.geometric_augmentation() function.

Take, for example, He/aug-cc-pVTZ:

BASIS "ao basis" SPHERICAL PRINT
#BASIS SET: (7s,3p,2d) -> [4s,3p,2d]
He    S
      2.340000E+02           2.587000E-03           0.000000E+00           0.000000E+00           0.00000000
      3.516000E+01           1.953300E-02           0.000000E+00           0.000000E+00           0.00000000
      7.989000E+00           9.099800E-02           0.000000E+00           0.000000E+00           0.00000000
      2.212000E+00           2.720500E-01           0.000000E+00           0.000000E+00           0.00000000
      6.669000E-01           4.780650E-01           1.000000E+00           0.000000E+00           0.00000000
      2.089000E-01           3.077370E-01           0.000000E+00           1.000000E+00           0.00000000
      0.0513800              0.00000000             0.00000000             0.00000000             1.0000000
He    P
      3.044000E+00           1.000000E+00           0.000000E+00           0.00000000
      7.580000E-01           0.000000E+00           1.000000E+00           0.00000000
      0.1993000              0.00000000             0.00000000             1.0000000
He    D
      1.965000E+00           1.0000000              0.00000000
      0.4592000              0.00000000             1.0000000
END

The parameters are generated as below:

	S	P	D
Smallest Exponents	2.089000E-01	7.580000E-01	1.965000E+00
	0.0513800	0.1993000	0.4592000
X	5.1380E-02	1.9930E-01	4.5920E-01
Y	2.089E-01	7.58E-01	1.965E+00
d-aug exponent	1.2637E-02	5.2402E-02	1.0731E-01
t-aug exponent	3.1082E-03	1.3778E-02	2.5077E-02
q-aug exponent	7.6447E-04	3.6226E-03	5.8603E-03

Therefore, He/q-aug-cc-pVTZ will be:

BASIS "ao basis" SPHERICAL PRINT
#BASIS SET: (7s,3p,2d) -> [4s,3p,2d]
He    S
      2.340000E+02           2.587000E-03           0.000000E+00           0.000000E+00           0.00000000
      3.516000E+01           1.953300E-02           0.000000E+00           0.000000E+00           0.00000000
      7.989000E+00           9.099800E-02           0.000000E+00           0.000000E+00           0.00000000
      2.212000E+00           2.720500E-01           0.000000E+00           0.000000E+00           0.00000000
      6.669000E-01           4.780650E-01           1.000000E+00           0.000000E+00           0.00000000
      2.089000E-01           3.077370E-01           0.000000E+00           1.000000E+00           0.00000000
      0.0513800              0.00000000             0.00000000             0.00000000             1.0000000
He    S
      1.264E-02              1.0
He    S
      3.108E-03              1.0
He    S
      7.645E-04              1.0
He    P
      3.044000E+00           1.000000E+00           0.000000E+00           0.00000000
      7.580000E-01           0.000000E+00           1.000000E+00           0.00000000
      0.1993000              0.00000000             0.00000000             1.0000000
He    P
      5.240E-02              1.0
He    P
      1.378E-02              1.0
He    P
      3.623E-03              1.0
He    D
      1.965000E+00           1.0000000              0.00000000
      0.4592000              0.00000000             1.0000000
He    D
      1.073E-01              1.0
He    D
      2.508E-02              1.0
He    D
      5.860E-03              1.