Notes about the Jensen basis sets
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admm basis sets
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The most widely used approach to approximate the Coulomb and exchange integrals is density fitting, also known as the resolution-of-the-identity (RI) approximation.
In this, the products of two one-electron basis functions are expanded in one-center auxiliary functions. RI significantly improves performance with a limited impact on
the accuracy and has therefore been applied to HF/KS theory as well as correlated methods. ADMM has been developed specifically for the exchange contribution. The exchange
energy is split into two parts. On consisting of the exact HF exchange and the second is a first-order correction term, evaluated as the difference between the generalized
gradient approximation exchange in the full and auxiliary basis sets.
aug-pc-n and pc-n basis sets (where n indicates the level of polarization beyond the atomic system)
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For the basis sets pc-0 through pc-4, the exponents were optimized by a pseudo-Newton-Raphson approach. Molecular geometries were either taken from experimental work
or MP2/cc-pVTZ optimized.
The pc-n basis sets have been designed to systematically improve the representation of the wave function, which is the primary contributor to energetic quantities.
It is also expected that the addition of diffuse functions (aug-pc-n basis sets) will improve the performance of the excitation energies.
The Diamagnetic Spin-Orbit Contribution is insensitive to the addiion of tight functions, as their influence is only indirect by changes in the density.
The Paramagnetic Spin-Orbit Contribution provides the largest contribution to the coupling constant
The Spin Dipole Contribution provides the second largest contribution to the coupling constant
Below are the DSO PSO and SD contributions (Hz) to the spin-spin coupling in F2 (which was used as a representative case)
DSO PSO SD
BASIS NONE +s +p +d +f BASIS NONE +s +p +d +f BASIS NONE +s +p +d +f
pc-0 3.41 3.41 3.04 pc-0 7032 7033 9497 pc-0 3733 3774 5172
pc-1 2.53 2.53 2.39 2.36 pc-1 8542 8542 9569 9565 pc-1 4805 4805 5406 5691
pc-2 2.20 2.20 2.17 2.14 2.13 pc-2 9245 9245 9418 9416 9409 pc-2 5426 5426 5527 5800 5926
pc-3 2.12 2.12 2.12 2.11 2.11 pc-3 9399 9399 9417 9417 9416 pc-3 5728 5738 5749 5873 5927
pc-4 2.11 2.11 2.11 2.11 2.11 pc-4 9409 9409 9414 9414 9414 pc-4 5865 5865 5867 5873 5911
aug-pc-0 3.30 3.30 2.99 aug-pc-0 7745 7746 10483 aug-pc-0 4341 4341 5969
aug-pc-1 2.49 2.49 2.36 2.33 aug-pc-1 8623 8623 9611 9658 aug-pc-1 4870 4870 5481 5770
aug-pc-2 2.20 2.20 2.17 2.14 2.13 aug-pc-2 9245 9245 9419 9416 9409 aug-pc-2 5428 5428 5530 5803 5929
aug-pc-3 2.12 2.12 2.12 2.11 2.11 aug-pc-3 9395 9295 9414 9413 9413 aug-pc-3 5737 5737 5748 5872 5926
pcJ-n basis sets
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Optimized for calculating indirect nuclear spin-spin coupling constants
using density functional methods. The calculation of nuclear magnetic spin-spin constats are a challenging task because the results largely depend on the quality of
the wave function. There are two limiting types of basis set contraction, denoted segmented and general. In a segmented contraction, each primitive basis function is
only allowed to contribute to one contracted function, while in a general contraction, all primitive functions are allowed to contribute to all contracted functions.
The contraction coefficients for a general contraction are normally obtained from SCF coefficients.
Jensen2010a performed a systematic search for the optimum contraction schemes for the pcJ-n sets and permformed extensive analysis on the s and p functions of atoms and
documented the results. The work provides guidelines for searching for optimum contraction schemes at theoretical levels, like CCSD, where a systematic search is impractical.
Below are the recommended contraction schemes for the pcJ-n Basis Sets:
Hydrogen 1st Row 2nd Row
Basis Uncontracted Contracted Uncontracted Contracted Uncontracted Contracted
pcJ-0 5s 3s 7s4p 4s3p 10s7p 5s4p
pcJ-1 6s2p 4s 9s5p2d 5s4p 13s9p2d 6s5p
pcJ-2 8s3p2d 5s 12s7p3d2f 7s5p 15s11p3d2f 8s6p
pcJ-3 11s5p3d1f 8s 16s10p5d2f1g 10s8p 19s14p5d3f1g 11s10p
pcJ-4 12s7p4d2f1g 10s 19s12p7d3f2g1h 15s10p 22s17p7d4f2g1h 16s12p
pcX-n basis sets
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The self-consistent-field method (SCF) includes orbital relaxation effects by separate optimizations of the ground and the core–hole states, and calculates the
CEBE (core-electron binding energy) as the difference between total electronic energies, and this method is widely used. The first SCF calculations were performed using HF
which ignores electron correlation and relativistic effects, but the inclusion of the relaxation energy allowed reproduction of experimental results. The SCF method can
be extended to post HF-methods such as MPn, MCSCF and CCSD methods, but they significantly increase the required computational resources and achieving convergence of the highly
excited hole state using coupled cluster methids can be difficult. PcX-n basis sets can approach the basis set limit for core-excitation processes such as XPS and XAS calculated
by the SCF approach. They are shown to provide lower basis set errors than other basis sets of similar size, and can be used as local basis sets in combination with the energy
optimized pcseg-n basis sets for reducing computational costs when core-excitation processes are only required for a small subset of atoms.
pcseg-n and psSseg-n basis sets
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These segmented basis sets were constructed from polarization consistent basis sets. An energy-optimized uncontracted basis set is augmented with property specific functions
to improve the basis set convergence. It is then general contracted based on the property basis set error relative to the complete basis set limit at each ζ level, and finally
converted into a computational more efficient segmented contraction by removing the redundancy among the primitive functions in the general contracted functions. The pcSseg-n
basis sets are in qualities ranging from double-ζ to pentuple-ζ quality and should be suitable for both routine and benchmark calculations of nuclear magnetic shielding constants.
While these basis sets have been developed for use with DFT methods, they may also be suitable for wave function methods including electron correlation. They can furthermore be
used as locally dense basis sets in combination with the energy-optimized pcseg-n basis sets for large systems, where shielding constants are only required for a small subset
of atoms.
pcH basis sets
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Note that these differ slightly from those given in the SI of jakobsen2019a. The data here contains slight corrections.
Notes from Frank Jensen
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pcseg: the most recent (2014) segmented contracted version of the pc-n basis sets, defined for H-Kr, for n=0,1,2,3,4
apcseg: the most recent (2014) segmented contracted version of the aug-pc-n basis sets, defined for H-Kr, for n=0,1,2,3,4
These should be used for DFT energetic properties (energies, geometry optimization, vibrational frequencies, ...)
The apcseg has in addition a full set of diffuse functions, useful for anions, polar systems, and electric properties (dipole moments, polarizabilities, ...)
NOTE: the old general contracted pc-n should be considered out-dated, as the pcseg have the same (or slightly better) performance and are computationally
more efficient
pcSseg: the most recent (2015) segmented contracted version of the pcS-n basis sets, defined for H-Kr, for n=0,1,2,3,4
apcSseg: the most recent (2015) segmented contracted version of the aug-pcS-n basis sets, defined for H-Kr, for n=0,1,2,3,4
These should be used for NMR shielding constants, and have been shown to be better than other basis sets also for correlated methods (MP2, CC)
The apcSseg has in addition a full set of diffuse functions, which may improve results for some systems
NOTE: the old general contracted pcS-n should be considered out-dated, as the pcSseg have the same (or slightly better) performance and are computationally
more efficient
pcJ: the most recent contracted version of the pcJ-n basis sets, defined for H-Ar, for n=0,1,2,3,4
apcJ: the most recent contracted version of the aug-pcJ-n basis sets, defined for H-Ar, for n=0,1,2,3,4
These should be used for NMR spin-spin coupling constants, and are probably also close to optimum for correlated methods (MP2, CC)
The apcJ has in addition a full set of diffuse functions, which may improve results for some systems
pcX: basis sets optimized for core-spectroscopy, defined for Li-Ar, for n=1,2,3,4
apcX: as pcX, with a full set of diffuse functions, defined for Li-Ar, for n=1,2,3,4
These should be used for X-ray spectroscopy (XPS) in combination with Delta-SCF methods, especially when using DFT, but are probably also suitable for
correlated methods (MP2, CC)
The apcX has in addition a full set of diffuse functions, which is important/essential for XAS
These basis sets are uncontracted, as any contraction destroys the accuracy. The n=1,2 basis sets are generated by exponent interpolation, while the n=3,4
simply are the uncontracted versions of the original (general contracted) pc-n basis sets, as no improvement was found by exponent interpolation
NOTE: these basis sets should not be used for XPS/XAS using response methods, as basis set requirements here are different than when using Delta-SCF methods.
Basis sets suitable for response methods are in progress.
admm: auxiliary basis sets for representing the exchange energy using the auxiliary-density matrix method (ADMM), defined for Li-Ar, for n=1,2,3
aadmm: as admm, with a full set of diffuse functions, defined for Li-Ar, for n=1,2,3
These basis sets are designed to be used in combination with pcseg-n and aug-pcseg-n, with the same n, when using the ADMM method in Dalton
Each of the directories contains the basis sets as files: ATOM-n.fmt, where fmt is a format for specific programs:
fmt = inp : Gaussian
fmt = gms : Gamess-US
fmt = mol : Dalton
fmt = mol2 : Dalton, old(er) versions
Key references:
pcseg: Atoms H-Kr:
F. Jensen "Unifying General and Segmented Contracted Basis Sets. Segmented Polarization Consistent Basis Sets." J. Chem. Theory Comp. 10 (2014) 1074-1085
These build on previous versions for selected subgroups of atoms using a general contraction:
F. Jensen "Polarization Consistent Basis Sets. Principles." J. Chem. Phys. 115 (2001) 9113-9125; 116 (2002) 3502.
F. Jensen "Polarization Consistent Basis Sets III. The Importance of Diffuse Functions." J. Chem. Phys. 117 (2002) 9234-9240.
F. Jensen, T. Helgaker "Polarization Consistent Basis Sets V. The Elements Si-Cl." J. Chem. Phys. 121 (2004) 3463-3470
F. Jensen "Polarization Consistent Basis Sets. VI. The Elements He, Li, Be, B, Ne, Na, Mg, Al, Ar." J. Phys. Chem. A 111 (2007) 11198-11204
F. Jensen "Polarization Consistent Basis Sets. VII. The Elements K, Ca, Ga, Ge, As, Se, Br and Kr." J. Chem. Phys. 136 (2012) 114107
F. Jensen "Polarization Consistent Basis Sets. VIII. The Transition Metals Sc-Zn." J. Chem. Phys. 138 (2013) 014107
pcSseg: Atoms H-Kr:
F. Jensen "Segmented Contracted Basis Sets Optimized for Nuclear Magnetic Shielding." J. Chem. Theory Comp. 11 (2015) 132-138
These build on previous versions for atoms H-Ar using a general contraction:
F. Jensen "Basis Set Convergence of Nuclear Magnetic Shielding Constants Calculated by Density Functional Methods." J. Chem. Theory Comp. 4 (2008) 719-727
pcJ: Atoms H,He,B-Ne,Al-Ar:
F. Jensen "The Optimum Contraction of Basis Sets for Calculating Spin-Spin Coupling Constants." Theor. Chem. Acc. 126 (2010) 371-382
The contractions for the Al-Ar pcJ-0 and pcJ-1 have subsequently been changed from general to segmented based on unpublished work.
These build on previous versions using a different (less) contraction:
F. Jensen "The Basis Set Convergence of Spin-Spin Coupling Constants Calculated by Density Functional Methods." J. Chem. Theory Comp. 2 (2006) 1360-1369
Atoms Li,Be,Na,Mg:
P. A. Aggelund, S. P. A. Sauer, F. Jensen "Development of polarization consistent basis sets for spin-spin coupling constant calculations for the atoms
Li, Be, Na and Mg." J. Chem. Phys. 149 (2018) 044117
pcX: Atoms Li-Ar:
M. A. Ambroise, F. Jensen "Probing Basis Set Requirements for Calculating Core Ionization and Core Excitation Spectroscopy by the Delta-SCF Approach"
J. Chem. Theory Comp. 14 (2018) 0000
admm: Atoms
C. Kumar, H. Fliegl, F. Jensen, A. M. Teale, S. Reine, T. Kjaergaard "Accelerating Kohn-Sham Response Theory using Density Fitting and the
Auxiliary-Density-Matrix Method." Int. J. Quant. Chem. 118 (2018) e25639
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REFERENCES MENTIONED ABOVE
(not necessarily references for the basis sets)
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jakobsen2019a
Jakobsen, Philip, Jensen, Frank
Probing basis set requirements for calculating hyperfine coupling
constants
J. Chem. Phys. 151, 174107 (2019)
10.1063/1.5128286