On the Ordering of Orbital Energies in the ROHF Method: Koopmans Theorem Versus Aufbau

On the ordering of orbital energies in the ROHF method: Koopmans’ theorem versus Aufbau principle

B. N. Plakhutin, A. V. Novikov, N. E. Polygalova, and T. E. Prokhorov

Supplementary material

Orbital-energy scaling technique

Here we present a description of a simple and effective orbital-energy scaling technique outlined in Refs. [1, 2], which enables one to perform ROHF computations of systems violating the Aufbau principle. In the below treatment we numerate the formulas by the numbers (S1), (S2), … , and use the original numbers (1), (2), … for the formulas from the main part of the paper. A list of references to this section is presented below. The orbital-energy scaling technique is based on Montgomery’s algorithm [3] of solving the ROHF equation for HSHFOS systems implemented in the GAMESS suite of programs [3, 4].

ˆ The ROHF diagonal operators R(ss) ( s  c , o , v ) entering the total ROHF Hamiltonian [Eqs. (16) and (19)] can be presented in two equivalent forms

ˆ ˆ ˆ ˆ R(ss)  2{ A(ss) Fo  B(ss) ( Fc  Fo )}

 A(ss) Fˆ  B(ss) Fˆ  , (S1) ˆ ˆ  where both the ROHF Fock operators, Fc and Fo (18), and the UHF Fock operators Fˆ and Fˆ  [5] are formed using the same ROHF densities. Based on this equivalence, the diagonal ˆ operators R(ss) are constructed in the algorithm [3] from the operators Fˆ and Fˆ  . The coefficients A(ss) and B(ss) of Eq. (S1) are specific for the ROHF method used (see also Table I in the main part of the paper). In the algorithm [3], these coefficients are used as input data, and this allows one to use the same algorithm [3] for solving the HF equation (7) with different ROHF Hamiltonians. Of particular importance is that the algorithm [3] can be used also for solving each of two uncoupled HF equations (35)-(36) appearing in the canonical ROHF method

ˆ  I II [6,7], since the diagonal operators R(ss) entering ROHF Hamiltonians Rˆ and Rˆ (35)-(36) take the same form (S1), (see also Table II).

1 If the ROHF orbital energies i violate the Aufbau principle (5), i.e.,

(k  m  v ),or(k  m  v ), (S2) it leads to known difficulties in the iterative SCF procedure. A simple idea to overcome these difficulties can be commented as follows. From the general ROHF scheme [Eqs. (6)-(16)] it

ˆ follows that a multiplication of any of three diagonal operators R(ss) in Eq. (19) on arbitrary nonzero number  s does not affect the ROHF orbitals and total energy. The latter statement becomes obvious, if we assume that the ROHF orbitals used to evaluate the matrix (19) obey the variational principle. For this case, the matrix (19) takes a block-diagonal form, in which the dia- ˆ gonal blocks are formed by matrix elements of operators R(ss) [see also Eq. (15) and the respec- ˆ tive text]. If any of the diagonal operators R(ss) in the block-diagonal form of (19) is multiplied ~ ˆ ˆ by arbitrary number s , the eigenvectors of the new operator R(ss)  R(ss)   s remain equal ˆ to those of the operator R(ss) , while the eigenvalues of these two operators are different. For ˆ example, if one multiplies the operator R(oo) by a nonzero number o , ~ ˆ ˆ R(oo)  R(oo)  o , (S3) ~ ˆ ˆ the eigenvectors of R(oo) are as the same as ones of R(oo) , while the eigenvalues of these two operators are connected by the relationship ~ m  m  o . (S4) ˆ ˆ The eigenvectors and eigenvalues of two other (unchanged) diagonal operators, R(cc) and R(vv) , remain unchanged. In those cases, where the ROHF orbital energies i violate the Aufbau prin- ˆ ciple (5) in any of two forms (S2), one should multiply the diagonal operator R(oo) by a nonzero number o (S3) such that ~ k  m  v . (S5) This relationship means a fulfillment of the Aufbau principle for the modified (temporary) orbital energies (S4). A fulfillment of (S5) in the iterative procedure lifts all difficulties caused by violation of the AP (5). After a convergence is achieved, the proper values of ROHF open-shell ~ orbital energies m are obtained by back transformation m  m /o . Within the algorithm [3], a multiplication (S3) corresponds to the use of the new coefficients in Eq. (S1) ~ ~ A(ss)  A(ss)  o , B(ss)  B(ss)  o , (S6)

2 and thus, does not require any modification of the algorithm [3].

Our experience has shown that the value of o in (S6) which ensures a fulfillment of the Aufbau principle (S5) in most open-shell systems is equal to 1/ 3. It is easy to see that in all cases of violation of the AP (5) presented in Tables III and IV of the main part of the paper, the multiplier o  1/ 3 provides a fulfillment of the AP (S5). For example, the canonical ROHF orbital energies  (7t1u ) and  (6hu ) for endofullerene N@ C60 equal to 15.701 and 7.595, respectively (see Table III), violate the AP (5). The respective modified orbital energies ~  (7t1u )  15.701/3 and  (6hu )  7.595 obey the AP (S5). In some specific cases, such as the molecules containing 3d -metal atoms, the recommended value of o is equal to 1/10 [7]. In all cases the use of small non-zero multipliers o  1/10 (or less) positively effects on the convergence of the iterative SCF procedure.

References

1. Plakhutin BN, Davidson ER (2009), J Phys Chem A 113:12386 2. Plakhutin BN, Davidson ER (2014), J Chem Phys 140:014102 3. http://www.msg.ameslab.gov/GAMESS/GAMESS.html 4. Schmidt MW, Baldridge KK, Boatz JA, Elbert ST, Gordon MS, Jensen JH, Koseki S, Matsunaga N, Nguyen KA, Su S, Windus TL, Dupuis M, Montgomery JA (1993), J Comput Chem 14:1347 5. Pople JA, Nesbet RK (1954), J Chem Phys 22:571 6. Plakhutin BN, Gorelik EV, Breslavskaya NN (2006), J Chem Phys 125:204110 7. Davidson ER, Plakhutin BN (2010), J Chem Phys 132:184110