Basis sets

What a basis set is

A basis set is the collection of functions used to express the molecular orbitals. Quantum-chemistry calculations project the exact Schrödinger equation onto a finite function space, so the choice of basis sets the upper limit on accuracy. Larger bases are more accurate but their cost grows as the 4th–6th power of system size.

Main families — def2, cc, ANO

Sizes — DZ, TZ, QZ

Labels like "double-zeta (DZ)" and "triple-zeta (TZ)" indicate how many functions describe each valence orbital. The presence of polarisation (d, f) and diffuse functions further determines the total size of the basis.

Diffuse functions

Diffuse functions (prefix aug- or ma-, or the + / ++ annotations) add large gaussians that extend far from the nucleus, describing loosely bound electrons. They are essential for:

• Anions: the negative charge loosely binds the extra electron.
• Excited states: especially Rydberg states (high-energy excitations).
• Polarisabilities and hyperpolarisabilities: response to large fields.
• Weak interactions: hydrogen bonds, π–π stacking, etc.

Auxiliary basis sets

RI acceleration, RIJCOSX and MP2 / CC calculations decompose four-index integrals into a "three-index + auxiliary basis" form, so an auxiliary basis must be specified alongside the main basis. A wrong auxiliary basis is a common source of accuracy loss, so this deserves attention.

Pairing the auxiliary basis correctly

# RIJCOSX hybrid DFT
! B3LYP D4 def2-TZVP RIJCOSX def2/J

# RI-JK hybrid DFT
! B3LYP D4 def2-TZVP RI-JK def2/JK

# DLPNO-MP2 or DLPNO-CCSD(T)
! DLPNO-MP2 cc-pVTZ cc-pVTZ/C

Effective core potentials (ECPs)

For heavy elements (broadly 4d onwards — element 36, Rb, and beyond) it is efficient to replace the explicit core electrons with an effective core potential, which also folds in scalar-relativistic effects. The def2 family automatically uses the def2-ECP from element 37 onwards.

# Silver (Ag) atom — def2-ECP is applied automatically
! B3LYP D4 def2-TZVP

* xyz 0 2
  Ag   0.0   0.0   0.0
*

For more precise treatment, use a scalar-relativistic basis (ZORA, DKH, X2C) instead, which describes all electrons explicitly:

# Ag atom — ZORA scalar relativity with the ZORA-recontracted basis
! B3LYP D4 ZORA ZORA-def2-TZVP SARC/J

* xyz 0 2
  Ag   0.0   0.0   0.0
*

SARC — for heavy elements

For 4d, 5d, 4f, and 5f elements (roughly elements 37, Rb, through 100, Fm) the SARC (Segmented All-electron Relativistically Contracted) basis is the recommended choice. It is accurate at the DFT level and pairs cleanly with ZORA / DKH scalar relativity.

# Gd(III) complex — 4f element
! BP86 ZORA SARC2-ZORA-QZVP SARC/J

* xyzfile 3 8 Gd_complex.xyz

Recommendations by situation

The importance of consistency

When comparing reaction energies or bond-dissociation energies, always apply the same basis to every species so the basis-set superposition errors (BSSE) cancel. BSSE partially cancels when the same basis is used, but the cancellation breaks down if bases are mixed.

BSSE and counterpoise correction — quantifying intermolecular interactions

When two molecules sit close to one another, each molecule's basis effectively "borrows" functions from the other's region, so the dimer energy comes out artificially low compared to the sum of the monomer energies. This is the Basis-Set Superposition Error (BSSE), and it is responsible for overestimating binding energies in hydrogen bonds, π–π stacks, and vdW complexes by 1–3 kcal/mol.

The textbook fix is the Boys–Bernardi counterpoise (CP) correction. In ORCA, a ghost atom is created by appending a colon : to the element symbol. A ghost atom carries a basis set but has no nucleus and no electrons — a "virtual" atom.

The 4-step CP procedure

Below is the calculation of the binding energy of a water-methanol dimer with the CP correction. The four jobs are chained inside a single input via $new_job.

# Step 1: full dimer — the "raw" binding energy
! B3LYP D4 def2-TZVP RIJCOSX def2/J TightSCF

* xyz 0 1
  O   0.0000   0.0000   0.0000        # water O
  H   0.0000   0.7572  -0.5800
  H   0.0000  -0.7572  -0.5800
  O   2.7800   0.0000   1.8400        # methanol O
  C   3.2400   0.0000   3.1500
  H   3.2400   1.0300   3.4900
  H   3.2400  -1.0300   3.4900
  H   4.2700   0.0000   2.7900
  H   2.5000   0.0000   1.3000
*

$new_job
# Step 2: water alone (methanol as ghosts) — A on the dimer basis
! B3LYP D4 def2-TZVP RIJCOSX def2/J TightSCF

* xyz 0 1
  O   0.0000   0.0000   0.0000
  H   0.0000   0.7572  -0.5800
  H   0.0000  -0.7572  -0.5800
  O:  2.7800   0.0000   1.8400        # ghost
  C:  3.2400   0.0000   3.1500
  H:  3.2400   1.0300   3.4900
  H:  3.2400  -1.0300   3.4900
  H:  4.2700   0.0000   2.7900
  H:  2.5000   0.0000   1.3000
*

$new_job
# Step 3: methanol alone (water as ghosts) — B on the dimer basis
! B3LYP D4 def2-TZVP RIJCOSX def2/J TightSCF

* xyz 0 1
  O:  0.0000   0.0000   0.0000
  H:  0.0000   0.7572  -0.5800
  H:  0.0000  -0.7572  -0.5800
  O   2.7800   0.0000   1.8400
  C   3.2400   0.0000   3.1500
  H   3.2400   1.0300   3.4900
  H   3.2400  -1.0300   3.4900
  H   4.2700   0.0000   2.7900
  H   2.5000   0.0000   1.3000
*

$new_job
# Step 4: reference — methanol on its own (monomer basis only, no ghosts)
! B3LYP D4 def2-TZVP RIJCOSX def2/J TightSCF

* xyz 0 1
  O   2.7800   0.0000   1.8400
  C   3.2400   0.0000   3.1500
  H   3.2400   1.0300   3.4900
  H   3.2400  -1.0300   3.4900
  H   4.2700   0.0000   2.7900
  H   2.5000   0.0000   1.3000
*

(A separate Step 4 for water alone on its own monomer basis is also required; for brevity only methanol is shown.) The four energies give two binding energies:

Other uses of ghost atoms

Ghost atoms (:) are useful beyond CP correction:

• Extending the basis of a neighbouring fragment: putting ghosts around an active site adds basis-function freedom without treating those atoms explicitly.
• Floating orbitals for polarisability: ghost atoms placed outside the molecule expand the basis available for response functions.
• Reference centres for EPR / NMR: explicitly fixing the origin when computing magnetic properties.