Post on 25-Jun-2018
transcript
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Computational Chemistry (KJE-3102) Basis Sets
Bin Gao (bin.gao@uit.no)
Center for Theoretical and Computational ChemistryDepartment of Chemistry
University of Troms
Feb. 11 and 21, 2011
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Outline
1 Generalities about Basis Sets
2 Slater Basis Sets
3 Gaussian Basis Sets
4 Integral Evaluation
5 Pseudopotentials
6 GEN1INT Tool Package
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 2 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Schrodinger Equation
Based on Born-Oppenheimer approximation, the motions of electrons can beseparated from nuclei, which results in the electronic Schrodinger equation
Hee = Eee, (1)
with
He = Te + Vne + Vee + Vnn (2)
= N
i=1
122i
N
i=1
M
A=1
ZAriA
+N1
i=1
N
j=i+1
1rij
+ Vnn. (3)
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 3 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Solving the Schrodinger EquationThe underlying physical laws necessary for the mathematical
theory of a large part of physics and the whole of chemistry are thuscompletely known, and the difficulty is only that the exact applicationof these laws leads to equations much too complicated to be soluble.
P. A. M. Dirac
The equationHee = Eee
is a many-body problem, too complicated to solve directly. Approximate waysof solution
Numerical methods, such as finite-difference, and finite-elementapproach.Expansion method represent e as
e(x1,x2, . . . ,xN)
i
Ci i (x1,x2, . . . ,xN). (4)
J. R. Chelikowsky et al., Phys. Rev. Lett. 72, 1240 (1994).J. R. Chelikowsky et al., Phys. Rev. B 50, 11355 (1994).J. E. Pask et al., Phys. Rev. B 59, 12352 (1999).
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 4 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Slater Determinants
Pauli exclusion principle the wave function e must be antisymmetricwith respect to the permutation of any two electrons, i.e.
e(x1,x2, . . . ,xi, . . . ,xj, . . . ,xN) = e(x1,x2, . . . ,xj, . . . ,xi, . . . ,xN). (5)
Slater determinant:
SD(x1,x2, . . . ,xN) =1N!
1(x1) 2(x1) N(x1)1(x2) 2(x2) N(x2)
......
. . ....
1(xN) 2(xN) N(xN)
, (6)
where i (x) is the molecular spin orbital, composed of spatial orbital andspin function,
i (x) = i (r)i (s), (i = , ). (7)
Note that it is one-electron wave function (from independent particle model)!Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 5 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Molecular Orbitals and Basis Sets
The spatial part i (r) could in principle be constructed by a linearcombination of basis functions:
i (r) =
j
cijj (r), (8)
where cij is the molecular orbital (MO) coefficient, {j (r)} in definition is acomplete basis set and would require an infinite number of basisfunctions. However, in practice, one has to use a finite number of basisfunctions, for instance the Gaussian Type Orbital (GTOs) function.Linear Combination of Atomic Orbitals (LCAO) strictly speaking,Atomic Orbitals (AO) are solutions for the atom. Atomic Basis Functionshould be more appropriate here.
They should be the basis functions of Hilbert space.Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 6 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Hartree-Fock-Roothaan Method
Converged!
Yes
No1 ?n nD D
1Determine nD
0Get initial density matrix D
Compute nF D
Figure: Illustration of SCF procedure.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 7 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
MOs from DALTON program
i (r) =
j
cijj (r)
Molecular orbitals for symmetry species 1------------------------------------------
Orbital 1 2 3 41 O :1s -1.0004 0.0070 -0.0000 -0.00032 O :1s -0.0013 -0.9097 -0.0000 -0.12773 O :1s 0.0020 0.0270 0.0000 0.22154 O :2px 0.0000 -0.0000 -0.0000 0.00005 O :2py -0.0039 0.2408 0.0000 -0.76866 O :2pz 0.0000 0.0000 -0.9133 0.0000... ... ... ... ... ...
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 8 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Atomic Basis Functions
Mathematically speaking, the basis set is the set of functions from whichthe wave function is constructed.Requirements of suitable basis functions
The basis should be designed such that it allows for an orderly andsystematic extension towards completeness with respect to one-electronsquare-integrable functions.The basis should allow for a rapid convergence to any atomic or molecularelectronic state, requiring only a few terms for a reasonably accuratedescription of molecular electron distributions.The functions should have an analytical form that allows for easymanipulation. In particular, all molecular integrals over these functionsshould be easy to evaluate. It is also desirable that the basis functions areorthogonal or at least that their nonorthogonality does not present problemsrelated to numerical instability.
Christopher J. Cramer, Essentials of Computational Chemistry: Theories and Models(Second Edition).Trygve Helgaker, Poul Jrgensen, and Jeppe Olsen, Molecular Electronic-Structure Theory.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 9 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Different kinds of Basis Functions
Slater Type Orbitals (ADF)Gaussian Type Orbitals (DALTON, DIRAC, GAUSSIAN,GAMESS USA, ...)Plane Wave Basis Functions (modelling extended (infinite) systems)Wavelet (MRChem)SIESTA program: Numerical atomic orbitals for linear-scalingcalculations, J. Junquera, O. Paz, D. Sanchez-Portal, and E. Artacho,Phys. Rev. B 64, 235111, (2001).... ...
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 10 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Slater Type Orbitals (STO)
,n,l,m(r , , ) = NYlm(, )rn1er , (9)
where N is a normalization constant, is called exponent. The r , , and are spherical coordinates, and Ylm is the angular momentum part (sphericalharmonic function describing shape).Pros and Cons:
The exponential dependence on the distance between the nucleus andelectron mirrors the exact orbitals for the hydrogen atom exponentialdecay, s-type Slaters have a nuclear cusp (discontinuous derivative).The STOs do not have any radial nodes; nodes in the radial part areintroduced by making linear combinations of STOs.Very difficult to compute three- and four-centre two-electron integrals.Used in some atomic codes and some molecular codes, for instance,Amsterdam Density Functional (ADF) package.
Frank Jensen, Introduction to Computational Chemistry: Chapter 5.Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 11 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Gaussian Type Orbitals (GTO)
Proposed by Boys in 1950,
Spherical-harmonic: ,n,l,m(r , , ) = NYlm(, )r2nl2er2, (10)
Cartesian: ,lx ,ly ,lz (x , y , z) = Nxlx y ly z lz er
2, (11)
where N is a normalization constant, is the exponent, and Ylm is thespherical harmonic function.Cons and Pros:
Instead of cusp, a GTO has a zero slope at the nucleus, so that GTOshave problems representing the proper behaviour near the nucleus.The GTO falls off too rapidly far from the nucleus compared with an STO,and the tail of the wave function is consequently represented poorly.Therefore, more GTOs are necessary for achieving a certain accuracycompared with STOs.However, integrals are more efficient to compute, most commonly used inmolecular codes.
Frank Jensen, Introduction to Computational Chemistry: Chapter 5.Do not be confused by the orbital, they are simply basis functions!
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 12 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Gaussian Primitives
In practice, all applications take the GTOs to be centred at the nuclei. Forinstance, a primitive Cartesian Gaussian centered at O takes the form:
,lx ,ly ,lz (xO , yO , zO) = N(x Ox )lx (y Oy )ly (z Oz)lz er2O . (12)
Another better candidate when evaluating geometric derivatives HermiteGaussian:
,lx ,ly ,lz (xO , yO , zO) = N(2)lxlylz
(
Ox
)lx ( Oy
)ly ( Oz
)lzer
2O , (13)
as proposed by Reine, Tellgren and Helgaker (Phys. Chem. Chem. Phys. 9(2007), 4771).
The definition of primitive will be given in Contracted Basis Sets part.Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 13 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The exponents
How are these Gaussian primitives derived?Gaussian primitives are usually obtained from quantum calculations onatoms, the exponents are typically varied until the lowest total energy ofthe atom is achieved.The exponents could be either optimized independently, or related toeach other by some equation, and parameters in this equation areoptimized, for example, the even-tempered or well-tempered basissets.These primitives from atomic calculations cannot accurately describedeformations of atomic orbitals in the molecule. Augmented functions,such as polarization functions and diffuse functions, are thereforeusually used.In molecular calculations, these Gaussian primitives have to becontracted, i.e., certain linear combinations of them will be used as basisfunctions which are sometimes called Contracted Gaussian TypeOrbitals (CGTO).
Simplified Introduction to Ab Initio Basis Sets. Terms and Notation atwww.ccl.net/cca/documents/basis-sets/basis.html
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 14 / 55
www.ccl.net/cca/documents/basis-sets/basis.html
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Even-Tempered Basis Sets
The optimization of basis function exponents becomes difficult when thebasis set becomes large, since the basis functions start to becomelinearly dependent and the energy becomes a very flat function of theexponents.Notice that the ratio between two successive optimized exponents isapproximately constant, the optimization with fixed ratio thus involvesonly two parameters for each type of basis function, independent of thesize of the basis. Such basis sets are even-tempered basis sets, with thei th exponent given as
i = i . (14)
Pros: easy to generate a sequence of basis sets that are guaranteed toconverge towards a complete basis.Cons: however, the convergence is somewhat slow, and an explicitlyoptimized basis set of a given size will usually give a better answer thanan even-tempered basis of the same size.
It was later discovered that the optimum and constants to a good approximation can bewritten as functions of the size of the basis set, M Frank Jensen, Introduction toComputational Chemistry: Chapter 5.Frank Jensen, Introduction to Computational Chemistry: Chapter 5.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 15 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Well-Tempered Basis Sets
The exponents in a well-tempered basis of size M are generatedaccording to
i = i1[
1 + (
iM
)]; i = 1,2, . . . ,M, (15)
the parameters , , and are optimized for each atom.Compared to even-tempered basis set, the well-tempered basis set hasfour parameters, and is thus capable of giving a better result for the samenumber of functions.More general parameterization proposed by Petersson et al., see.From the point of view of users, we need to pick up a suitable basis setduring calculations, in particular, when our interest is in specializedproperties.
Frank Jensen, Introduction to Computational Chemistry: Chapter 5.Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 16 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Contracted Basis Sets
Contracted GTOs (CGTOs):
Spherical-harmonic: n,l,m(rO , O , O) = Ylm(O , O)r2nl2O
i
wiei r2, (16)
Cartesian: lx ,ly ,lz (xO , yO , zO) = xlxOy
lyOz
lzO
i
wiei r2, (17)
where the normalization constants of individual GTOs (known as the primitiveGTOs, PGTOs) have been adsorbed into the contraction coefficients wi .
(10s4p1d/4s1p)[3s2p1d/2s1p], or (10s4p1d/4s1p)/[3s2p1d/2s1p]: Thebasis in parenthesis is the number of primitives with heavy atoms (firstrow elements) before the slash and hydrogen after. The basis in thesquare brackets is the number of contracted functions.
Frank Jensen, Introduction to Computational Chemistry: Chapter 5.Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 17 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Segmented Contraction
STO-3G in DALTON:
$ CARBON (6S,3P) -> [2S,1P]$ S-TYPE FUNCTIONS
6 2 071.6168370 0.15432897 0.0000000013.0450960 0.53532814 0.000000003.5305122 0.44463454 0.000000002.9412494 0.00000000 -0.099967230.6834831 0.00000000 0.399512830.2222899 0.00000000 0.70011547
$ P-TYPE FUNCTIONS... ... ... ...
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 18 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
General Contraction
cc-pVDZ in DALTON:
$ CARBON (9s,4p,1d) -> [3s,2p,1d]$ S-TYPE FUNCTIONS
9 3 06665.0000000 0.00069200 -0.00014600 0.000000001000.0000000 0.00532900 -0.00115400 0.00000000228.0000000 0.02707700 -0.00572500 0.0000000064.7100000 0.10171800 -0.02331200 0.0000000021.0600000 0.27474000 -0.06395500 0.000000007.4950000 0.44856400 -0.14998100 0.000000002.7970000 0.28507400 -0.12726200 0.000000000.5215000 0.01520400 0.54452900 0.000000000.1596000 -0.00319100 0.58049600 1.00000000
$ P-TYPE FUNCTIONS... ... ... ...$ D-TYPE FUNCTIONS... ... ... ...
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 19 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Classification of Basis Sets
STO-3G (minimal basis, single- basis sets): proposed by Hehre,Stewart, and Pople in 1969 mimicking STOs with contracted GTOs.Double- basis sets: two functions for each AO size of basis set doesnot change, but the size of the secular equation would be increased.Triple- basis sets, and higher multiple- basis sets.Split-valence or valence-multiple- basis sets: core orbitals continueto be represented by a single (contracted) basis function, while valenceorbitals are split into arbitrarily many functions, for instance, 3-21G,6-21G, 4-31G, 6-31G, and 6-311G (split-valence basis sets usingsegmented contractions).cc-pVDZ and cc-pVTZ: correlation-consistent polarized Valence(Double/Triple) Zeta basis sets (split-valence basis sets using generalcontractions).
There is one and only one basis function defined for each type of orbital core throughvalence, other minimal basis sets include the MINI sets of Huzinaga and co-workers.Correlation-consistent implies that the exponents and contraction coefficients were
variationally optimized not only for HF calculations, but also for calculations including electroncorrelation.Christopher J. Cramer, Essentials of Computational Chemistry: Theories and Models
(Second Edition).Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 20 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Poples Basis Sets
n-ijG or n-ijkG:n is the number of primitives for the inner shells.ij or ijk are the number of primitives for contractions in the valence shell,and ij describes sets of valence double zeta quality and ijk sets ofvalence triple zeta quality.
Simplified Introduction to Ab Initio Basis Sets. Terms and Notation atwww.ccl.net/cca/documents/basis-sets/basis.html
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 21 / 55
www.ccl.net/cca/documents/basis-sets/basis.html
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Polarization Functions
Adding higher angular momentum basis functions (than any occupiedatomic orbital) to describe the polarization of the charge distribution uponformation of a chemical bond.6-31G* implies a set of d functions are added on first-row atoms.6-31G** implies except for a set of d functions added on first-row atoms,p functions are also added on hydrogen atom.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 22 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Diffuse Functions
The highest energy MOs of anions, highly excited electronic states, andloose supermolecular complexes, tend to be much more spatially diffusethan garden-variety MOs.Diffuse functions (functions with small exponents, hence large radialextent) are thus usually included.6-31+G(d): added one s and one set of p functions having smallexponents to heavy atoms.6-311++G(3df,2pd): the second plus indicates the presence of diffuse sfunctions on hydrogen atoms.
Christopher J. Cramer, Essentials of Computational Chemistry: Theories and Models(Second Edition)
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 23 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Extra Topic: Basis-set Linear Dependence
Basis sets become linearly dependent when the size of the basis and thesize of the molecule grow, in particular if the basis contains diffusefunctions.Orthonormalization of a linearly dependent basis set becomesproblematic.Remove MOs in DALTON:
*ORBITAL INPUT.AO DELETETHROVL
THROVL is the limit for basis set numerical linear dependence[eigenvectors (of AO overlap matrix) with eigenvalue less than THROVLare excluded]. Default is 106.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 24 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Extra Topic: Basis-set Superposition Error (BSSE)
Molecular interaction energy EAB (EA + EB).A systematic error would occur when using finite basis sets Basisfunctions from A can help compensate for the basis set incompletenesson B, and vice versa. The energy of AB will therefore be artificiallylowered, and the interaction energy will be overestimated. This is knownas Basis-set Superposition Error (BSSE).Using more basis functions requires very large basis sets, not feasible.Counterpoise-corrected interaction energy EAB (ECPA + ECPB ), whereECPA is the counterpoise-corrected energy of A calculated in the full basisof AB, i.e., by including the ghost basis functions on B. Likewise for ECPB .
Trygve Helgaker, Poul Jrgensen, and Jeppe Olsen, Molecular Electronic-Structure TheoryBin Gao (CTCC, UiT) Computational Chemistry KJE-3102 25 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Extra Topic: Crystal Orbitals
k ,(r) =1Nsite
g
eik g(r g), (18)
|k , PBC =1Nsite
g
eik g |g, AO. (19)
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 26 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Extra Topic: London Atomic Orbitals (LAO)
The London Atomic Orbital is defined by multiplying the conventional AOs bya field-dependent phase factor
(r) = exp[ i2 B (R G)(r P)
]
(r R)(|r R|), (20)
where is centered at the nuclear position R, B is the magnetic field, G isthe gauge origin of the magnetic vector potential, and P the origin of theLondon phase factor exp
[ i2 B (R G)(r P)
]. The second factor
(r) in eqn (20) is the angular part of the AO, typically a solid harmonicfunction Sl,m(r) in the position r of the electron relative to the AO. Finally,(r) is a decaying radial form in r, usually chosen as a contracted Gaussian
(r) =
i
wi exp(air2), (21)
where wi and ai are the radial contraction coefficients and orbitalexponents, respectively.Bast et al., Phys. Chem. Chem. Phys. 13 (2011), 2627.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 27 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Recommended Literatures and Websites
Christopher J. Cramer, Essentials of Computational Chemistry: Theoriesand Models (Second Edition): Chapters 4.3 and 6.2.Frank Jensen, Introduction to Computational Chemistry: Chapter 5.Trygve Helgaker, Poul Jrgensen, and Jeppe Olsen, MolecularElectronic-Structure Theory: Chapters 6, 8 and 9.Simplified Introduction to Ab Initio Basis Sets. Terms and Notation atwww.ccl.net/cca/documents/basis-sets/basis.html
EMSL Basis Set Exchange at https://bse.pnl.gov/bse/portalSegmented Gaussian Basis Set athttp://setani.sci.hokudai.ac.jp/sapporo/Welcome.do
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 28 / 55
www.ccl.net/cca/documents/basis-sets/basis.htmlhttps://bse.pnl.gov/bse/portalhttp://setani.sci.hokudai.ac.jp/sapporo/Welcome.do
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
One- and Two-electron Integrals
O = |O1| =(r)O(r)(r)dr , (22)
O = (1)(1)|O12|(2)(2) (23)
=
(r1)(r1)O(r1, r2)(r2)(r2)dr1dr2.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 29 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Gaussian Product Rule
The product of two Gaussians can be written in terms of another Gaussianfunctions, for instance,
exp(ax2A) exp(bx2B) = exp(X 2AB) exp(px2P), (24)
where
p = a + b total exponent, (25)
=ab
a + breduced exponent, (26)
Px =aAx + bBx
pcenter-of-charge coordinate, (27)
XAB = Ax Bx relative coordinate/Gaussian separation, (28)K xab = exp(X 2AB) pre-exponential factor. (29)
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 30 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Gaussian Overlap Distribution
ab(r) = Gikm(r ,a,A)Gjln(r ,b,B) (30)
= xij (x ,a,b,Ax ,Bx )ykl (y ,a,b,Ay ,By )
zmn(z,a,b,Az ,Bz), (31)
where the x component is
xij (x ,a,b,Ax ,Bx ) = Gi (x ,a,Ax )Gj (x ,b,Bx ) (32)
= K xabxiAx
jB exp(px2P) (33)
= K xab
i+j
k=0
C ijk xkP exp(px2P). (34)
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 31 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Properties of Gaussian Overlap Distribution
xAxij = xi+1,j , (35)
xBxij = xi,j+1, (36)
xi,j+1 xi+1,j = XABxij , (37)xijAx
= 2axi+1,j ixi1,j , (38)
xijBx
= 2bxi,j+1 jxi,j1, (39)(Px
= Ax
+ Bx
XAB
= bpAx ap
Bx
(40)
xijPx
= 2axi+1,j + 2bxi,j+1 ixi1,j jxi,j1, (41)
xijXAB
= 2(xi+1,j xi,j+1) +1
2p(2ajxi,j1 2bixi1,j ). (42)
Trygve Helgaker, Poul Jrgensen, and Jeppe Olsen, Molecular Electronic-Structure Theory:Chapter 9.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 32 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Integrals of Primitives Obara-Saika Scheme
Overlap integrals Sab = Ga|Gb = SijSklSmn, where
Sij =Z
xij dx . (43)
Translational recurrence relation
2aSi+1,j iSi1,j + 2bSi,j+1 jSi,j1 = 0, (44)
and horizontal recurrence relation
Si,j+1 Si+1,j = XABSij , (45)
Obara-Saika Scheme
Si+1,j = XPASij +1
2p(iSi1,j + jSi,j1), (46)
Si,j+1 = XPBSij +1
2p(iSi1,j + jSi,j1), (47)
and starting from
S00 =r
pexp(X 2AB). (48)
Trygve Helgaker, Poul Jrgensen, and Jeppe Olsen, Molecular Electronic-Structure Theory:Chapter 9.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 33 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Integrals of Primitives McMurchie-DavidsonScheme
Hermite Gaussians tuv (r ,p,P) = (/Px )t (/Py )u(/Pz)v exp(pr2P).Overlap distribution expanded in Hermite Gaussians
ij =
i+j
t=0
E ijt t , (49)
t (x ,p,Px ) = (/Px )t exp(pr2P). (50)Auxiliary distributions
tij = Kxabx
iAx
jBt (xP), (51)
and the McMurchie-Davidson recurrence relations
ti+1,j = tt1ij + XPA
tij +
12p
t+1ij , (52)
ti,j+1 = tt1ij + XPB
tij +
12p
t+1ij . (53)
Trygve Helgaker, Poul Jrgensen, and Jeppe Olsen, Molecular Electronic-Structure Theory:Chapter 9.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 34 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Coulomb Integrals
The complications arise from the presence of the inverse operator 1rC or1
r12.
Using the fact that
1rC
=1
+
exp(r2C t2)dt , (54)
the key-quantity in the evaluation of Coulomb integrals is the so-calledBoys function
Fn(x) = 1
0exp(xt2)t2ndt (55)
and its recurrence relations.
Trygve Helgaker, Poul Jrgensen, and Jeppe Olsen, Molecular Electronic-Structure Theory:Chapter 9.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 35 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Other Schemes of Evaluating Integrals
Gaussian quadrature see, for instance, Trygve Helgaker, PoulJrgensen, and Jeppe Olsen, Molecular Electronic-Structure Theory:Chapter 9.6.The multiple method for Coulomb integrals see, for instance, TrygveHelgaker, Poul Jrgensen, and Jeppe Olsen, MolecularElectronic-Structure Theory: Chapter 9.13, and Elias Rudberg and PaweSaek, J Chem. Phys. 125 (2006), 084106.Cholesky decomposition for instance, I. Reggen and Tor Johansen,J. Chem. Phys. 128 (2008), 194107, and Linus Boman et al., J. Chem.Phys. 129 (2008), 134107.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 36 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Integrals of Contracted GTOs
(r) = (r R)
i
wi exp(air2), (56)
O = |O1| =(r)O(r)(r)dr (57)
=
ij
wiwj(r R) exp(air2)O(r)(r R) exp(ajr2 )dr .
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 37 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Symmetry
One-electron operator
O = |O1| = |O1| = O (symmetric), (58)
or
O = |O1| = |O1| = O (anti-symmetric). (59)
Two-electron integral O = (1)(1)|O12|(2)(2)
O = O = O = O (60)= O = O = O = O.
From group theory, the integral can only be none zero if the direct productof the bra and ket irreps is equal to (or contains) an irrep correspondingto the operator.
Christopher J. Cramer, Essentials of Computational Chemistry: Theories and Models(Second Edition): Appendix B.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 38 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Effective Core Potentials (ECP)
The ECP operator around the core C with L 1 being the largest angular momentum orbital
U(rC) = UL(rC) +L1Xl=0
lXm=l
|Ylm[Ul (rC) UL(rC)]Ylm|, (61)
The functions Ylm(C , C) are real spherical harmonics centered on C
Ylm(C , C) =
8>>>:Y 0l (C , C), if m = 0,
12
hY ml (C , C) + (1)
mYml (C , C)i, if m > 0,
1i
2
hYml (C , C) (1)
mY ml (C , C)i, if m < 0,
(62)
where Y ml is the complex spherical harmonics.The radial functions UL(rC) and Ul (rC) UL(rC) (l = 0, . . . , L 1) are expressed ascombinations of Gaussians
UL(rC) =Ncore
rC+
Xk
dkLrnkLC eckLr2C , (63)
Ul (rC) UL(rC) =X
k
dkl rnklC eckl r2C , (64)
where Ncore is the number of core electrons. nkL and nkl are generally restricted to 0, 1, and 2.Chris-Kriton Skylaris et al. Chem. Phys. Lett. 296 (1998) 445451.Larry E. McMurchie and Ernest R. Davidson, J. Comp. Phys. 44 (1981) 289301.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 39 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Model Core Potentials (MCP)
In MCP Version 1, for an atom C with the atomic charge Z and Ncore core electrons,the interaction operator V core between core and valence electrons is approximated as
V core Vmp(rC) = Z Ncore
rC
Xl
Al rnlC el r2C , (65)
where nl = 0 or 1, and the parameters Al and l are adjustable parameters of MCP.Moreover, in order to prevent the valence orbitals from collapsing onto those of coreelectrons, the so-called energy shift operator is usually added into the final Hamiltonian
= X
k
fkk |k (rC) k (rC)| , (66)
where 1 fk 2 are adjustable parameters, and k is the eigenvalue of the k th coreorbital. k (rC) is the k th core orbital, which is represented by real solid-harmonicGaussian functions.
Sigeru Huzinaga, Can. J. Chem. 73 (1995) 619628.Mariusz Klobukowski et al., in Computational Chemistry: Reviews of Current Trends, 3 (1999)
4974.Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 40 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
MCP (Continue)
In MCP Version 2 (AIMP), the Coulomb potential is represented as
V coreC V coreC =X
j
Aj r nj1 exp(j r 2), (67)
while the exchange operator V coreX is used in the spectral representation VcoreX
V coreX = PVcoreX P (68)
whereP =
Xpq
|p(S1)pqp| (69)
is defined using a finite, non-orthogonal basis {p} with the metric Spq = p|q. Thecore projector operator still has the form
= 2X
k
k |k k |. (70)
Version III of the model potential method is the logical extension of the AIMP method,in which the entire core operator V core = V coreC + V
coreX is used in the spectral
representationV core V core = P(V coreC + V coreX )P. (71)
Sigeru Huzinaga, Can. J. Chem. 73 (1995) 619628.Mariusz Klobukowski et al., in Computational Chemistry: Reviews of Current Trends, 3 (1999)
4974.Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 41 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Extra Topic: Integrals between Crystal Orbitals
A(k) = k , |A|k , PBC =1
Nsite
g,h
eik (hg)g, |A|h, AO
=
l
eik l0, |A|l , AO. (72)
Challenge: evaluation of Coulomb integrals.See, for example, references at CRYSTAL program homepagehttp://www.crystal.unito.it/.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 42 / 55
http://www.crystal.unito.it/
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
What is GEN1INT?
GEN1INT is a Fortran 90 library (with Python interface) to evaluate thegeometric and magnetic derivatives of one-electron integrals
at zero field (for instance B = 0), andusing contracted London atomic orbitals (LAO)(r ; B) = exp
[ i2 B (R G) rP
](r).
More explicitly, what we evaluate is
N
LR
[K 1B
(r ; B)
]B=0
O({rC} ,nr
) [K 2B (r ; B)
]B=0
dr (73)
wherethe number of differentiated centers N 4, so thatthe number of centers in operator O satisfies N 2,the operator O may also depend on the magnetic field B, which givesnew operator
[
K OB O
]B=0
.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 43 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
A Generalized One-electron Operator
O({rC} ,nr
)=
Cf ({rC}) nr ,Effective core potential,Model core potential (Version 1).
(74)
where
f ({rC}) =
rmM[LCC r
m0C
], (m0 = 1,2),
rmM[LCC (rC)
],
rmM ,
LC1C1
LC2C2
r1C1 r1C2,
erf(%rC)
rC.
(75)
Multi-index notations
rmM = xmxM y
myM z
mzM , (76)
nr =
(
x
)nx ( y
)ny ( z
)nz. (77)
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 44 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Geometric Derivatives
Take a molecule with 12 atoms as an example, the number of third orderone-center geometric derivatives is 120, and 2376 for the two-center geometricderivatives, 5940 the three-centers 8436 third order geometric derivatives intotal.What is worse ;-) some guys has special requirement about the order ofgeometric derivatives
1(1)
1(1)
0
2(2)
1
3(2)
1
1(1)
0
2(2)
1
3(2)
1
2(2)
0
3(3)
1
3(2)
0
1(1)
0
2(2)
1
3(2)
1
2(2)
0
3(3)
1
3(2)
0
2(2)
0
3(3)
1
3(3)
0
3(2)
0
H = 3L H + 1 = 4
1 1 |L1|=2
2 2 |L2|=2
(|L1|+ 22
)
X1
X1
X1
Y1
X1
Z1
Y1
Y1
Y1
Z1
Z1
Z1
(|L2|+ 22
)
X2
X2
X2
Y2
X2
Z2
Y2
Y2
Y2
Z2
Z2
Z2
K=2
k=1
(|Lk|+ 22
)
X1
X1
X2
X2
X1
X1
X2
Y2
Z1
Z1
Z2
Z2
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 45 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Geometric Derivatives (Continue)
X1
R1(r)O ({rC} ,nr )1(r)dr (78)
=R h
X1
1(r)i
O1(r)dr +R1(r)O
hX1
1(r)i
dr .
0 (translational invariance).
Table: Possible geometric derivatives for R = R and the operator with the number ofcenters N 2.
Coincide centers Possible geometric derivativesN = 0 R = R 0N = 1 R = R = C1 0
R = R 6= C1 (1)|L|LC1 +LC1
N = 2 R = R = C1 = C2 0
R = R = C1 6= C2 (1)|L|LC2 +LC2
(R = R) 6= (C1 = C2) (1)|L|PL+LC1
l1=0
`L+LC1l1
l1C1
L+LC1l1C2
R = R 6= C1 6= C2PL
l=0
`Ll
lR
LlR
LC1C1
LC2C2
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 46 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Magnetic Derivatives
In order to evaluate
LRLR
[K 1B
(r ; B)
]B=0
O({rC} ,nr
) [K 2B (r ; B)
]B=0
dr (79)
to any order K 1 and K 2, we introduce the following auxiliary integral
LRLR
rN1P
[K 1B
(r ; B)
]B=0
O({rC} ,nr
)rN2P
[K 2B (r ; B)
]B=0
dr , (80)
and we have the recurrence relations
{K 1+e,LN1}K 0L0 = i2[(RG)+1{K 1L,N1+e1}K 0L0 (81) (RG)1{K 1L,N1+e+1}K 0L0+ (L)+1{K 1,Le+1,N1+e1}K 0L0 (L)1{K 1,Le1,N1+e+1}K 0L0
],
likewise for K 2.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 47 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Magnetic Derivatives (Continue)
(a) the order of functions in shells
do nz = 0, Kmaxdo ny = 0, Kmaxnz
(Kmax(ny+nz), ny, nz)end do
end do
0 y
x
z
yy
xy
xx
xz
yz
zz
xxz
yyy
xyy
xxy
xxx
xyz
yyz
xzz
yzz
zzz
xyyz
xxyz
xxxz
yyyy
xyyy
xxyy
xxxy
xxxx
yyyz
xxzz
xyzz
yyzz
xzzz
yzzz
zzzz
Kmax = 5
yyyyz
xyyyz
xxyyz
xxxyz
xxxxz
yyyyy
xyyyy
xxyyy
xxxyy
xxxxy
xxxxx
xxxzz
xxyzz
xyyzz
yyyzz
xxzzz
xyzzz
yyzzz
xzzzz
yzzzz
zzzzz
{x}{y}{z}
{x}
{x}{y}
{x}{y}{z}
{x}
{x}
{x}
{y}{x}
{x}{y}
{x}{y}
{z}
{x}
{x}
{x}
{x}
{y}{x}
{x}
{x}{y}
{x}
{x}{y}
{x}{y}
{z}
{x}
{x}
{x}
{x}
{x}
{y}{x}
{x}
{x}
{x}
{y}{x}
{x}
{x}{y}
{x}
{x}{y}
{x}{y}
{z}
(b) recurrence relations
{K1+e,LN1}K0= i2
[(RG)+1{K1L,N1+e1}K0
(RG)1{K1L,N1+e+1}K0+ (L)+1{K1,Le+1,N1+e1}K0 (L)1{K1,Le1,N1+e+1}K0
],
do k = Kmax, 1, 1do nx = 1, k
recurrence relation along x directionend dorecurrence relation along y direction
end dorecurrence relation along z direction
This is also the order of basisfunctions (in shells), and geo-metric derivatives in GEN1INT.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 48 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Class Organization of GEN1INT
+contracted_cgto()+contracted_sgto()+contracted_lcgto()+contracted_lsgto()
ContrInt
-Center of mass-Gauge origin-Dipole origin-Origin of London phase factor
+__init__()+set_center_of_mass()+set_gauge_origina()+set_dipole_origin()+set_london_orgin()+overlap()+diplen()+potenergy()
PropCGTO
-Center of mass-Gauge origin-Dipole origin-Origin of London phase factor
+__init__()+set_center_of_mass()+set_gauge_origina()+set_dipole_origin()+set_london_orgin()+overlap()+diplen()+potenergy()
PropSGTO
>>> import Gen1Int.PropCGTO>>> S = Gen1Int.PropCGTO.overlap \. . . (coord_bra, exp_bra, coeff_bra, \. . . coord_ket, exp_ket, coeff_ket)
User
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 49 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
ContrInt
contractedCGTO
contractedSGTO
geometricderivatives
contractedLondon SGTO
contractedLondon CGTO
contractedHGTO
contractedLondon HGTO
rmM
[LCC r
2C
]nrr
mM
[LCC r
1C
]nrr
mM
[LCC (rC)
]nrr
mM
nr
erf(%rC)rC
nrMCP
(Version 1) ECP LC1C1
LC2C2
r1C1 r1C2
nr
recurrencerelations quadrature
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 50 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Parallelization
1(1)
1(1)
0
2(2)
1
3(2)
1
1(1)
0
2(2)
1
3(2)
1
2(2)
0
3(3)
1
3(2)
0
1(1)
0
2(2)
1
3(2)
1
2(2)
0
3(3)
1
3(2)
0
2(2)
0
3(3)
1
3(3)
0
3(2)
0
H = 3L H + 1 = 4
1 1 |L1|=2
2 2 |L2|=2
(|L1|+ 22
)
X1
X1
X1
Y1
X1
Z1
Y1
Y1
Y1
Z1
Z1
Z1
(|L2|+ 22
)
X2
X2
X2
Y2
X2
Z2
Y2
Y2
Y2
Z2
Z2
Z2
K=2
k=1
(|Lk|+ 22
)
X1
X1
X2
X2
X1
X1
X2
Y2
Z1
Z1
Z2
Z2
basis functions of ket
basi
sfu
ncti
ons
ofbr
a
a processor
mpi4py (MPI for Python - Python bindings for MPI) for parallelization.Parallel IO using HDF5 (h5py in Python).
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 51 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Status of GEN1INT
Not available for the time being :-(http://sourceforge.net/projects/gen1int (stable tar ball),http://repo.ctcc.no/projects/gen1int (developing version).Up to four-center geometric derivatives.The results of high order derivatives and/or large orbital quantumnumbers may not be reliable for, such as nuclear-attraction integrals,which is due to the evaluation of Boys functions.
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 52 / 55
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Recommended Literatures and Websites
Christopher J. Cramer, Essentials of Computational Chemistry: Theoriesand Models (Second Edition): Chapters 4.3 and 6.2.Frank Jensen, Introduction to Computational Chemistry: Chapter 5.Trygve Helgaker, Poul Jrgensen, and Jeppe Olsen, MolecularElectronic-Structure Theory: Chapters 6, 8 and 9.EMSL Basis Set Exchange at https://bse.pnl.gov/bse/portalSegmented Gaussian Basis Set athttp://setani.sci.hokudai.ac.jp/sapporo/Welcome.do
Pseudopotentials of the Stuttgart/Cologne group athttp://www.theochem.uni-stuttgart.de/pseudopotentials/index.en.html
The Ab Initio Model Potential Library athttp://www.uam.es/departamentos/ciencias/quimica//aimp/Data/AIMPLibs.html
Two-electron integral library Libint at http://www.files.chem.vt.edu/chem-dept/valeev/software/libint/libint.html
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 53 / 55
https://bse.pnl.gov/bse/portalhttp://setani.sci.hokudai.ac.jp/sapporo/Welcome.dohttp://www.theochem.uni-stuttgart.de/pseudopotentials/index.en.htmlhttp://www.theochem.uni-stuttgart.de/pseudopotentials/index.en.htmlhttp://www.uam.es/departamentos/ciencias/quimica//aimp/Data/AIMPLibs.htmlhttp://www.uam.es/departamentos/ciencias/quimica//aimp/Data/AIMPLibs.htmlhttp://www.files.chem.vt.edu/chem-dept/valeev/software/libint/libint.htmlhttp://www.files.chem.vt.edu/chem-dept/valeev/software/libint/libint.html
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Thank You for Your Attention !!
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
The Centre for Theoretical and Computational Chemistry
Exercise 7 of KJE-3102: Basis Sets 2
Please use any program language you are familiar with, implement theObara-Saika scheme of overlap integrals between two Cartesian Gaussians(along x direction)
Si+1,j = XPASij +1
2p(iSi1,j + jSi,j1), (82)
Si,j+1 = XPBSij +1
2p(iSi1,j + jSi,j1), (83)
starting from
S00 =
pexp(X 2AB). (84)
Please report your code and plot the overlap integrals using i = j = 2,a = b = 1.0, and XAB = 0.2,0.4,0.6,0.8,1.0,1.5,2.0,3.0,4.0,6.0,8.0,10.0.What conclusion could you draw from this picture?
Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 55 / 55
Generalities about Basis SetsSlater Basis SetsGaussian Basis SetsIntegral EvaluationPseudopotentialsGen1Int Tool Package