and valence atomic orbitals. The use of more basis functions is motivated by a desire to provide additional variational flexibility so the lCAO process can generate molecular orbitals of variable diffuseness as the local electronegativity of the atom varies 3. A triple-zeta(tz)basis in which three times as many CGtOs are used as the number of core and valence atomic orbitals(of course, there are quadruple-zeta and higher-zeta bases also) Optimization of the orbital exponents(Cs or a's )and the gto-to-CGTO contraction coefficients for the kind of bases described above have undergone explosive growth in recent years. The theory group at the Pacific Northwest National Labs(Pnnl) offer a world wide web site from which one can find(and even download in a form prepared for input to any of several commonly used electronic structure codes)a wide variety of Gaussian atomic basis sets. This site can be accessed at http://www.emsl.pnlgov:2080/forms/basisform.html c. Polarization functions One usually enhances any core and valence basis set with a set of so-called polarization functions. They are functions of one higher angular momentum than appears in the atom s valence orbital space(e.g, d-functions for C, N, and O and p-functions for H), and they have exponents(C or a)which cause their radial sizes to be similar to the sizes of the valence orbitals(i. e, the polarization p orbitals of the H atom are similar in size to the ls orbital ) Thus, they are not orbitals which describe the atom's valence orbital with one higher l-value; such higher-l valence orbitals would be radially more diffuse
11 and valence atomic orbitals. The use of more basis functions is motivated by a desire to provide additional variational flexibility so the LCAO process can generate molecular orbitals of variable diffuseness as the local electronegativity of the atom varies. 3. A triple-zeta (TZ) basis in which three times as many CGTOs are used as the number of core and valence atomic orbitals (of course, there are quadruple-zeta and higher-zeta bases also). Optimization of the orbital exponents (z’s or a's) and the GTO-to-CGTO contraction coefficients for the kind of bases described above have undergone explosive growth in recent years. The theory group at the Pacific Northwest National Labs (PNNL) offer a world wide web site from which one can find (and even download in a form prepared for input to any of several commonly used electronic structure codes) a wide variety of Gaussian atomic basis sets. This site can be accessed at http://www.emsl.pnl.gov:2080/forms/basisform.html. c. Polarization Functions One usually enhances any core and valence basis set with a set of so-called polarization functions. They are functions of one higher angular momentum than appears in the atom's valence orbital space (e.g, d-functions for C, N, and O and p-functions for H), and they have exponents (z or a) which cause their radial sizes to be similar to the sizes of the valence orbitals ( i.e., the polarization p orbitals of the H atom are similar in size to the 1s orbital). Thus, they are not orbitals which describe the atom's valence orbital with one higher l-value; such higher-l valence orbitals would be radially more diffuse
The primary purpose of polarization functions is to give additional angular flexibility to the LCAO process in forming bonding orbitals between pairs of valence atomic orbitals. This is illustrated in Fig 6.2 where polarization d orbitals on C and o are seen to contribute to formation of the bonding orbital of a carbonyl group by allowing polarization of the carbon atoms p orbital toward the right and of the oxygen tom' s p orbital toward the left
12 The primary purpose of polarization functions is to give additional angular flexibility to the LCAO process in forming bonding orbitals between pairs of valence atomic orbitals. This is illustrated in Fig. 6.2 where polarization dp orbitals on C and O are seen to contribute to formation of the bonding p orbital of a carbonyl group by allowing polarization of the carbon atom's pp orbital toward the right and of the oxygen atom's pp orbital toward the left
Carbon P and d orbitals combining to form Oxygen P, and d orbitals combining to form a bent r orbital I bond formed from C ando bent (polarized) AOs Figure 6.2 Oxygen and Carbon Form a T Bond That Uses the Polarization Functions on Each Atom Polarization functions are essential in strained ring compounds because they provide the angular flexibility needed to direct the electron density into regions between bonded atoms, but they are also important in unstrained compounds when high accuracy is required
13 Figure 6.2 Oxygen and Carbon Form a p Bond That Uses the Polarization Functions on Each Atom Polarization functions are essential in strained ring compounds because they provide the angular flexibility needed to direct the electron density into regions between bonded atoms, but they are also important in unstrained compounds when high accuracy is required. C O C O C O C O C O Carbon pp and dp orbitals combining to form a bent p orbital Oxygen pp and dp orbitals combining to form a bent p orbital p bond formed from C and O bent (polarized) AOs
d. diffuse functions When dealing with anions or Rydberg states, one must further augment the Ao basis set by adding so-called diffuse basis orbitals. The valence and polarization functions described above do not provide enough radial flexibility to adequately describe either of these cases. The PNNl web site data base cited above offers a good source for obtaining diffuse functions appropriate to a variety of atoms Once one has specified an atomic orbital basis for each atom in the molecule, the LCAO-MO procedure can be used to determine the Cui coefficients that describe the occupied and virtual (i.e, unoccupied)orbitals. It is important to keep in mind that th basis orbitals are not themselves the SCF orbitals of the isolated atoms; even the proper atomic orbitals are combinations(with atomic values for the Cui coefficients)of the basis functions. The LCAO-MO-SCF process itself determines the magnitudes and signs of the C.:. In particular, it is alternations in the signs of these coefficients allow radial nodes to 4. The Hartree-Fock Apprxoimation Unfortunately, the Hartree approximation discussed above ignores an important property of electronic wave functions-their permutational antisymmetry. The full Hamiltonian H=2i+/2m V:-Ze7r, ) +1/2 Eik e7lr-rkl
14 d. Diffuse Functions When dealing with anions or Rydberg states, one must further augment the AO basis set by adding so-called diffuse basis orbitals. The valence and polarization functions described above do not provide enough radial flexibility to adequately describe either of these cases. The PNNL web site data base cited above offers a good source for obtaining diffuse functions appropriate to a variety of atoms. Once one has specified an atomic orbital basis for each atom in the molecule, the LCAO-MO procedure can be used to determine the Cm,i coefficients that describe the occupied and virtual (i.e., unoccupied) orbitals. It is important to keep in mind that the basis orbitals are not themselves the SCF orbitals of the isolated atoms; even the proper atomic orbitals are combinations (with atomic values for the Cm,i coefficients) of the basis functions. The LCAO-MO-SCF process itself determines the magnitudes and signs of the Cn,i . In particular, it is alternations in the signs of these coefficients allow radial nodes to form. 4. The Hartree-Fock Apprxoimation Unfortunately, the Hartree approximation discsussed above ignores an important property of electronic wave functions- their permutational antisymmetry. The full Hamiltonian H = Sj {- h2 /2m Ñ 2 j - Ze2 /rj} + 1/2 Sj,k e2 /|rj -rk |
is invariant (i.e, is left unchanged) under the operation Pii in which a pair of electrons have their labels (i,D permuted. We say that H commutes with the permutation operator Pi This fact implies that any solution to HY=EY must also be an eigenfunction of Pi Because permutation operators are idempotent, which means that if one applies P twice, one obtains the identity PP=l, it can be seen that the eigenvalues of P must be either +1 or-1. That is. if pp=cp then Ppp=cc P but pp= 1 means that cc= 1. soc=+l or As a result of h commuting with electron permutation operators and of the idempotency of P, the eigenfunctions Y must either be odd or even under the application of any such permutation Particles whose wave functions are even under P are called Bose particles or Bosons, those for which P is odd are called Fermions. Electrons belong to the latter class of particles The simple spin-orbital product function used in Hartree theory does not have the proper permutational symmetry. For example, the Be atom function Y= 1sa(1)lsB(2)2sa(3)2sB(4)is not odd under the interchange of the labels of electrons'3 and 4; instead one obtains lsa(1)lsB(2)2sa(4)2sB(3). However, such products of spin-orbitals (i.e, orbitals multiplied by a or B spin functions)can be made into properly antisymmetric functions by forming the determinant of an NxN matrix whose row index labels the spin orbital and whose column index labels the electrons For example, the Be atom function 1sa(1)IsB(2)2sa(3)2sB(4)produces the 4x4 matrix
15 is invariant (i.e., is left unchanged) under the operation Pi,j in which a pair of electrons have their labels (i, j) permuted. We say that H commutes with the permutation operator Pi,j. This fact implies that any solution Y to HY = EY must also be an eigenfunction of Pi,j Because permutation operators are idempotent, which means that if one applies P twice, one obtains the identity P P = 1, it can be seen that the eigenvalues of P must be either +1 or –1. That is, if PY = cY, then P P Y = cc Y, but PP = 1 means that cc = 1, so c = +1 or –1. As a result of H commuting with electron permutation operators and of the idempotency of P, the eigenfunctions Y must either be odd or even under the application of any such permutation. Particles whose wave functions are even under P are called Bose particles or Bosons,; those for which Y is odd are called Fermions. Electrons belong to the latter class of particles. The simple spin-orbital product function used in Hartree theory Y = Pk=1,N fk does not have the proper permutational symmetry. For example, the Be atom function Y = 1sa(1) 1sb(2) 2sa(3) 2sb(4) is not odd under the interchange of the labels of electrons’3 and 4; instead one obtains 1sa(1) 1sb(2) 2sa(4) 2sb(3). However, such products of spin-orbitals (i.e., orbitals multiplied by a or b spin functions) can be made into properly antisymmetric functions by forming the determinant of an NxN matrix whose row index labels the spin orbital and whose column index labels the electrons. For example, the Be atom function 1sa(1) 1sb(2) 2sa(3) 2sb(4) produces the 4x4 matrix