Single point energies
Single point energies are the simplest properties one might aim to obtain, they are the lowest energy solution for the Schrödinger equation. However, there are different methods to compute those, with varying quality levels:
Hartree-Fock (HF)
It is one of the first approximations developed [Slater1951], which does not include the so-called dynamic or even static correlation. To run an example HF single point calculation, simply use:
!HF DEF2-SVP
* xyz 0 1
O -3.56626 1.77639 0.00000
H -2.59626 1.77639 0.00000
H -3.88959 1.36040 -0.81444
*
on your main input, and the basis DEF2-SVP will be assigned to each atom on your list. The total energy will be printed in the end of the file, e.g.:
------------------------- --------------------
FINAL SINGLE POINT ENERGY -75.959335138038
------------------------- --------------------
It is one of the oldest and simpler methods, and it is not in general recommended if you need good energies. However, it is the basis of more complete methods such as CCSD and CASSCF.
Unrestricted HF (UHF)
If one chooses any multiplicity other than one, say three for a triplet, an UHF calculation is performed. In this particular case, one should always check the spin contamination of the results.
Warning
If the expectation value of the \(S^2\) (written as <S**2>) operator differs significantly from the ideal value, it might be that your system could only be treated with a multi-reference calculation such as CASSCF.
In the case of the water above as a triplet, one gets:
----------------------
UHF SPIN CONTAMINATION
----------------------
Expectation value of <S**2> : 2.005700
Ideal value S*(S+1) for S=1.0 : 2.000000
Deviation : 0.005700
which shows a minimal deviation and is thus suitable for use here.
Acceleration of the SCF (RIJDX and RIJCOSX)
One way to accelerate the solution of the SCF equations, necessary to compute the energy, is to use the RI option to compute the Coulomb integrals [Neese2003]. In that case, an auxiliary basis is also necessary. For the DEF2 family, the DEF2/J is set by default:
!HF DEF2-SVP DEF2/J RIJDX
If the basis chosen has no corresponding auxiliary basis, you can include de AUTOAUX flag for an automatic generation of this auxiliary basis:
!HF 6-31+G(d,p) AUTOAUX RIJDX
For an even faster speed-up, the COSX algorithm [Helmich-Paris2021] can be used together with the RI to compute the exchange integrals efficiently.
Density functional theory (DFT)
DFT energies can be obtained in a similar fashion, by choosing any functional from the ones available. For instance:
!B3LYP DEF2-SVP
* xyz 0 1
O -3.56626 1.77639 0.00000
H -2.59626 1.77639 0.00000
H -3.88959 1.36040 -0.81444
*
requires a calculation using the B3LYP [Becke1993], [Parr1988] functional. The calculated energy is:
------------------------- --------------------
FINAL SINGLE POINT ENERGY -76.320253607017
------------------------- --------------------
The same argument for checking spin-contamination applies here. Always have in mind that the DFT results may change depending on the functional, so one should check before using these and never compare energies from different methods!
ORCA 5 is fully compatible with functionals from LibXC (https://www.tddft.org/programs/libxc/). For a complete list of functionals and more details, please check the ORCA manual
Note
The same approximations to accelerate the SCF, the RIJDX and RIJCOSX described above can be used with DFT. Of course, the RIJCOSX only makes sense when using a hybrid functional, where HF exchange is also calculated.
Important
The RIJDX approximation is the default for non-hybrid functionals and RIJCOSX the default for hybrid ones. To turn these of, you can use !NORI or !NOCOSX.
MP2 perturbation theory
MP2 is a post Hartree-Fock method, which means it starts with HF as a basis and tries to include its missing dynamic correlation. It is implemented in ORCA in a very efficient way using the RI approximation or even in a pair-natural variant (DLPNO) that is extremely efficient [Neese2015], both can be called using:
!RI-MP2 cc-pVTZ cc-pVTZ/C
or:
!DLPNO-MP2 cc-pVTZ cc-pVTZ/C
Because it is a so called correlated method, it needs a special basis "/C" for the RI part. These are not available for all basis sets, but can be obtain through AUTOAUX:
!DLPNO-MP2 6-311++G(2d,2p) AUTOAUX
* xyz 0 1
O -3.56626 1.77639 0.00000
H -2.59626 1.77639 0.00000
H -3.88959 1.36040 -0.81444
*
You can check the correlation energy contribution as:
------------------------------------------------------
DLPNO-MP2 CORRELATION ENERGY: -0.240752652556 Eh
------------------------------------------------------
and the final energy is printed as usual.
Coupled Cluster (CC)
Coupled cluster calculations, in particular the CCSD(T) variant, are the gold standard for single point energies. However, in their usual formulation, they are also extremely costly from a computational point of view.
ORCA features a DLPNO variant that is much more efficient and presents almost linear scaling growth for larger systems [Neese2016], [Neese2013a], [Neese2013b]. In order to use that, simply set DLPNO-CCSD(T) in your input and choose an appropriate basis (also needs a "/C" basis or AUTOAUX)
!DLPNO-CCSD(T) cc-pVTZ cc-pVTZ/C
* xyz 0 1
O -3.56626 1.77639 0.00000
H -2.59626 1.77639 0.00000
H -3.88959 1.36040 -0.81444
*
and the CC results are printed as:
----------------------
COUPLED CLUSTER ENERGY
----------------------
E(0) ... -76.055469404
E(CORR)(strong-pairs) ... -0.267934936
E(CORR)(weak-pairs) ... -0.000104148
E(CORR)(corrected) ... -0.268039083
E(TOT) ... -76.323508487
Singles Norm <S|S>**1/2 ... 0.018648176
T1 diagnostic ... 0.006593126
with the final single point energy at the end:
------------------------- --------------------
FINAL SINGLE POINT ENERGY -76.330949199540
------------------------- --------------------
Please always check the T1 diagnostic value printed. A rule of thumb says, that for a value of the diagnostic of larger than 0.02 the results are not to be trusted and the HF reference might be poor.
There are many different coupled cluster and coupled pair methods that can be use in ORCA, check the documentation for more details.
Note
The RIJCOSX algorithm can be used to significantly accelerate MP2 and CC calculations!