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Calculation of vertical ionization potentials with the Piris natural orbital functional
Journal of Molecular Structure: THEOCHEM 770 (2006) 45–49 www.elsevier.com/locate/theochem
Calculation of vertical ionization potentials with the Piris natural orbital functional Pavel Leiva, Mario Piris * Institute of Physical and Theoretical Chemistry, Friedrich-Alexander-University Erlangen-Nuremberg, Egerlandstrasse 3, 91058 Erlangen, Germany Received 10 January 2006; received in revised form 1 March 2006; accepted 2 May 2006 Available online 10 May 2006
Abstract The Piris natural orbital functional (PNOF) based on a new approach for the two-electron cummulant has been used to predict vertical ionization potentials of 15 molecules. The ionization energies have been calculated using the extended Koopmans’ theorem. The calculated PNOF values are in good agreement with the corresponding experimental data. q 2006 Elsevier B.V. All rights reserved. Keywords: Ionization potentials; Extended Koopman’s theorem; Natural orbital functional
1. Introduction The theoretical investigation in quantum chemistry of natural orbital functional theory (NOFT) [1–10] has increased quickly in the last several years [11–42]. In this formalism, the energy functional of the one-particle reduced density matrix (1-RDM) is expressed in terms of the natural orbitals and their occupation numbers. A great advantage is that the kinetic energy is explicitly defined and the NOFT therefore does not have to invoke the concept of a fictitious noninteracting particle system. Moreover, NOFT incorporates fractional occupation numbers in a natural way, which provides a correct description of both dynamical and nondynamical correlation. Since, the electronic energy is given exactly in terms of the spinless 1-RDM G and the two-particle charge density r2, one can attain a NOFT using a reconstruction functional r2[G]. Such reconstruction functional in terms of two symmetric matrices, D and L, has been recently proposed by Piris [40]. The suggested form of these matrices (as functions of the natural occupation numbers) produces a natural orbital functional (NOF) that reduces to the exact expression for the total energy in two-electron systems [16,43]. One can generalize it to the N-electron systems, except for the off-diagonal elements of D. Alternatively, the mean value theorem and the * Corresponding author. Tel.: C49 9131 8527347; fax: C49 9131 8527736. E-mail address: [email protected] (M. Piris).
0166-1280/$ - see front matter q 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2006.05.001
partial sum rule for matrix D provide a prescription for deriving a practical NOF. The comparison with other theoretical methods showed that Piris NOF (PNOF) provides total energies and dipole moments close to accurate ab initio methods like CCSD(T). Prediction of ionization potentials (IPs) has been a frequent area of application in quantum chemistry [44–48]. Theoretical chemistry provides several alternatives to the calculation of IPs. First, IPs can be computed at any level of theory as the energy differences between the N and (NK1)-particle systems (DE). The energy-difference calculations can be avoided using the electron propagator theory [49], the equation-of-motion method [50], the one-particle Green’s function method [51] or the extended Koopmans’ theorem (EKT) [52,53]. The formalism of EKT not only describes the electron detachment in terms of one-electron quantities, but it also offers the advantage of being highly efficient from the computational point of view. IPs of a number of molecules have been computed from EKT formalism with different approximate electron correlation treatments. Comparison between EKT IP’s and DE IP’s indicate that EKT is exact for the first IP and capable of yielding highly accurate higher IPs [54,55]. The aim of this paper is to apply PNOF to the determination of IPs. This is not the first NOF study of ionization potentials via EKT in molecular systems. Pernal et al. [38] have recently determined the IPs of four selected molecules using NOFs proposed in Refs. [21,28,37]. The present study complements their work: it extends the range of systems to 15 molecules and also it considers the PNOF for the prediction of this property. The performance of the PNOF is established by carrying out a
46
P. Leiva, M. Piris / Journal of Molecular Structure: THEOCHEM 770 (2006) 45–49
statistical analysis of the IP relative absolute errors with respect to the experiment values.
sum rule for off-diagonal elements of matrix D (see Eq. (30) in Ref. [40]) was used. Inserting this expression into Eq. (7), one obtains:
2. Theory
EZ
0 X X ð2ni hii C n2i Kii Þ C ð2nj ni Jij KLji Kij Þ ij
i
We briefly describe here the theoretical framework of our approach. A more detailed description of the PNOF can be found in Ref. [40]. The electronic energy functional for N-electron systems is: X X X E Z2 ni hii C 2 ðnj ni KDji ÞJij K Lji Kij (1) i
ij
if i Z j Z 1
(2)
pffiffiffiffiffiffiffiffi Lji Z ½1K2qð0:5Knj Þqð0:5Kni Þ nj ni ;
if i sj
(3)
where q(x) is the unit step function also known as the Heaviside function. Unfortunately, DjiZnjni violates the obtained sum rule for this matrix in the general case of NO2. This implies that the functional form of nondiagonal elements of D remains unknown for N-electron systems. In contrast, one can assume that the diagonal elements of D equal the square of the occupation numbers Dii Z n2i
(4)
and the functional form (2) and (3) of matrix L can be readily generalized to Lii ZKni
if i Z j
(5)
pffiffiffiffiffiffiffiffi Lji Z ½1K2qð0:5Knj Þqð0:5Kni Þ nj ni ;
if i sj
(6)
By taking into account Eqs. (4) and (5), the energy functional (1) can be expressed as EZ
X
ni ð2hii C Kii Þ C 2
0 X
ðnj ni KDji ÞJij K
ij
i
0 X
Lji Kij
(7)
ij
in which the primes indicates that the iZj term is omitted. It is not evident how to approach Dji, for jsi, in terms of the occupation numbers. Due to this fact, the energy term in Eq. (7), which involves Dji is rewritten as 0 X ij
Dji Jij Z
X i
Ji
0 X j
Dji Z
X
Ji ni ð1Kni Þ
(8)
i
where Ji denotes the mean value of the Coulomb interactions Jij for a given orbital i taking over all orbitals jsi. Here, the
X
ni ð1Kni ÞðKii K2Ji Þ
(9)
i
A further simplification is accomplished by setting Ji zKii =2, which produces:
ij
The states {ji(r)} constitute a complete orthonormal set of single-particle real wave functions, where i denotes the orbital, and {ni} are their occupations. hii is the matrix element of the kinetic energy and nuclear attraction terms. JijZhijjiji and KijZ hijjiji are the electronic repulsion integrals. Note that if DjiZ0 and LjiZnjni, then our reconstruction functional yields the Hartree–Fock (HF) case. From the requirement that for any two-electron system (NZ2) the expression (1) should yield the exact energy functional of Ref. [16], one easily deduces that DjiZnjni and L11 ZKn1
C
EZ
0 X X ð2ni hii C n2i Kii Þ C ð2nj ni Jij KLji Kij Þ i
(10)
ij
The NOF (10) turns out to be identical with the selfinteraction-corrected Hartree functional proposed by Goedecker and Umrigar [16] except for the choice of phases given by the sign of Lji. There are also similarities between this practical NOF and the recently proposed BBC functional [37]. Since, both NOFs derive from the exact two-electron case, matrix L given by Eq. (6) agrees with the first repulsive correction (C1) introduced by Gritsenko et al. to restore the positive phase of the cross products between weakly occupied orbitals. The negative phase keeps otherwise in both functionals. However, the functional form of L is different for many other cross products. In the case of BBC functional the natural orbitals are divided into the groups of strongly occupied (occ), weakly occupied (virt) and frontier (frn) orbitals. Moreover, an orbital may be of the virt and frn type at the same time. If isj and (i,j2occ)n(i2occ, j2frn)n(j2occ, i2frn) then LjiZnjni differing from (6). Another important difference is the diagonal term (iZj), which coincides only if i;frn. The functional form (6) for the matrix elements of L between orbitals with occupation numbers larger than 0.5 gives a wrong description for the lowest occupied levels. The occupation numbers for these levels are identically equal to one. In order to ensure that these occupation numbers only are close to unity an additional term was introduced [40] in the functional form of Lji: pffiffiffiffiffiffiffiffi Lji Z ½1K2qð0:5Knj Þqð0:5Kni Þ nj ni C qðnj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (11) K0:5Þqðni K0:5Þ ð1Knj Þð1Kni Þ; if i sj One has to minimize the energy (10) both with respect to {ji} and {ni}. Bounds on the occupation numbers (0%ni%1) are enforced by setting ni Z ðsin gi Þ2 and varying the {gi} without constraints. Introducing the Lagrange multiplier m with the normalization condition, and the set of symmetric multipliers {3ji} with orthonormality constraints on the real orbitals, the functional whose extremum we seek is given by: " # X X 2 U Z EKm 2 ðsin gi Þ KN K2 3ji ½hji jjj iKdji (12) i
ij
P. Leiva, M. Piris / Journal of Molecular Structure: THEOCHEM 770 (2006) 45–49
The Euler equations for the independent variables {ji} and {gi} are, respectively ! X dU ^ i ni K Z 4 Vj jj 3ji Z 0 (13) dji j dU dE Z sinð2gi Þ K2m Z 0 dgi dni where the one-particle operator V^ is X L~ ij K1 1 ^ C ^ ^ Vð1Þ Z hð1Þ 2nj hjjrK jjiK hjjr jji P 12 12 12 ni j
(14)
(15)
In Eq. (15), the P^ 12 operator permutes electrons 1 and 2. Note that the sum includes the jZi term since we have ~ with diagonal elements equal n2i introduced a new matrix L and off-diagonal elements given by L through Eq. (11). In explicitly dependent on the 1-RDM NOFT [1], the variational derivative of the functional E with respect to the occupation numbers is equal to 2Vii. Moreover, the 1-RDM and the matrix of orbital energies, 3, may be simultaneously brought to diagonal form by the same unitary transformation [3]. In this case, Eq. (14) implies that operator V^ has an essentially degenerate eigenvalue spectrum, i.e. all the natural orbital eigenvalues are the same (m), and are equal to minus the vertical IP [10,56]. As a result, the chemical potential is given by the ratio: mZ
ei ni
(16)
Conversely, we have here an implicitly dependent on the 1-RDM NOFT, so the energy functional still depends on the 2-RDM. As it was pointed out by Donnelly [4], there is a fundamental difference between energy functionals based on 1and on 2-RDMs. Our energy functional (10) is not invariant with respect to a unitary transformation of the orbitals hence Eq. (13) for the optimum orbitals is actually the Lo¨wdin’s equation [57]. In general, this equation cannot be reduced to an eigenvalue problem diagonalizing the matrix 3, although by slight manipulation V^ can be transformed into a Hermitian operator with a nondegenerate spectrum of eigenvalues n [4]. Such construction is provided by the extension of Koopmans’ theorem [52,53]. The equation for the extended Koopmans’ theorem for ionization potentials may be derived by expressing the wavefunction of the (NK1)-electron system as a simple linear combination: X jJNK1 i Z Ci a^ i jJN i (17) i
In Eq. (17), a^i is the annihilation operator for an electron in orbital i, jJNi is the wavefunction of the N-electron system, (JNK1i is the wavefunction of the (NK1)-electron system and {Ci} are a set of coefficients to be determined. Optimizing the energy of the state JNK1 with respect to the parameters {Ci} and subtracting the energy of JN, gives the EKT equations as a
47
generalized eigenvalue problem FC Z nGC
(18)
where n are the EKT ionization potentials. In Eq. (18), the metric matrix is G with the occupation numbers {ni} along the diagonal and zeros in off-diagonal elements, and the transition matrix elements are given by D E ^ a^i jJN ZKVji ni ZK3ji Fji Z JN ja^ †j ½H; (19) This equation can be transformed by canonical orthonormalization using GK1/2. With this transformation Eq. (18) can be written as: F 0 C 0 Z nC 0
(20)
It is now clear from Eq. (19) that the diagonalization of the matrix n with the elements eji (21) nji ZKpffiffiffiffiffiffiffiffi nj ni yields ionization potentials as eigenvalues [4,38]. 3. Results and discussion The simplest treatment for determining vertical IPs is based on Koopmans’ theorem (KT), which states that the IP is given by the HF orbital energy with opposite sign (K3i), calculated in the neutral system [58]. This approach ignores the relaxation of the molecular orbitals after the ionizations. However, it has long recognized that in general there is an excellent agreement between the KT values and the experimental ones because of the fortuitous cancellation of the correlation and relaxation effects. Table 1 lists the obtained vertical IPs by PNOF–EKT together with KT IPs and experimental values. The IPs were calculated using the contracted Gaussian basis set 6-31G** [59], at the near-experimental geometries of neutral molecules given in Ref. [60]. A survey of this table reveals that KT, due to the neglect of orbital relaxation in cationic states, consistently overestimates the first IPs (FIPs), except for C2H2, HCN, Li2 and P2. The prevailing trend is that the values decrease in moving from KT to PNOF–EKT, and then from PNOF– EKT to experimental data. Conversely, in case of C2H2, Li2 and P2, the FIPs increase in moving from KT to PNOF– EKT, and then from PNOF–EKT to experimental values. For CH4 and HCN molecules, the PNOF–EKT IPs are greater than the KT and experimental results, whereas in case of HCl and SiO the PNOF–EKT FIPs are smaller than the other values. Generally, the PNOF–EKT FIPs move closer to experimental data. In fact, the relative error of the PNOF–EKT values is smaller than the relative error of KT ones. The KT and PNOF–EKT methods show relative absolute difference with respect to the experiment of 5.6 and 3.5%, respectively. Table 1 lists also the higher IPs (HIPs) calculated via KT and PNOF–EKT methods for the 15 molecules. In general, the
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P. Leiva, M. Piris / Journal of Molecular Structure: THEOCHEM 770 (2006) 45–49
Table 1 Vertical ionization potential in electron volt Molecule
MO
Koopmansa
PNOF– EKTb
Expc
C2H2
pu sg su t2 a1 s p s pg pu sg p s b2 b1 a1 b2 b1 a1 b2 p s s p s s s s sg pu su a1 e pu sg s p
10.86 18.27 20.73 14.75 25.58 14.90 17.22 21.67 17.88 21.82 20.19 17.06 20.23 11.87 14.46 17.58 18.72 13.53 15.56 19.10 12.93 17.00 30.42 13.32 15.63 22.01 4.84 8.16 17.13 16.63 21.11 11.44 16.87 10.14 11.14 11.77 12.76
11.12 18.24 20.12 14.86 25.14 14.59 17.28 21.58 16.74 21.22 22.21 16.82 19.99 11.54 14.63 17.44 18.40 13.08 15.54 18.93 12.51 16.73 26.57 13.66 15.67 21.64 4.90 8.00 16.74 17.14 20.96 11.05 16.93 10.39 10.63 11.41 12.21
11.49 16.70 18.70 14.40 23.00 14.01 16.85 19.78 15.87d 18.80d 21.10d 16.19 19.90 10.90 14.50 16.10 17.00 12.78 14.83 18.72 12.77e 16.60e 25.80e 13.61f 14.01f 19.86f 5.14g 7.70h 15.60 16.68 18.78 10.80 16.80 10.65i 10.84i 11.61j 12.19j
CH4 CO
F2
FH H2CO
H2O
HCl
HCN HCN Li2 LiH N2
NH3 P2 SiO
relative errors are slightly higher for HIPs (5.5%) than the corresponding for FIPs (3.5%). We find also this trend for the KT results (7.0% for HIPs and 5.6% for FIPs). Important cases are the F2 and N2 molecules. It is wellknown that the KT sg and pu IPs are in the wrong order for both molecules (see Table 1). PNOF–EKT calculations on F2 and N2 give valence shell IPs, which are in the correct order, and in general a numerical improvement is obtained over KT IPs. In case of the orbital pu for N2, the KT IPs is closer to experimental data but due to the mentioned wrong ordering. The PNOF–EKT and KT values (including all IPs) show relative errors with respect to the experiment of 4.7 and 6.5%, respectively. In general, the results are in good agreement with the corresponding experimental vertical IPs considering the small basis sets used for these calculations. 4. Conclusions The recent obtained PNOF [40] was used to determine vertical ionization potentials for a set of 15 selected molecules at the experimental geometries. The IPs obtained from PNOF–EKT method are compared with the experimental IPs. The agreement between our PNOF– EKT IPs and those obtained experimentally is somewhat less for higher IPs than for the first IPs. The PNOF–EKT method under study provide an improvement in the IPs over the KT approach. In general, we observed that the overall trends with the introduction of the electronic correlation by PNOF are satisfactory. The reliability of the PNOF in the prediction of the studied property has been illustrated. It is confirmed that EKT calculations produce IPs of comparable accuracy but at a substantially lower computational cost. Acknowledgements
a
K3HF i . b Vertical ionization potentials obtained from the extended Koopmans’ theorem. c Experimental vertical ionization potentials from Ref. [54]. d Experimental vertical ionization potential for F2 from Ref. [61]. e Experimental vertical ionization potential for HCl from Ref. [62]. f Experimental vertical ionization potential for HCN from Ref. [63]. g Experimental vertical ionization potential for Li2 from Ref. [64]. h Experimental adiabatic ionization potential for LiH from Ref. [65]. i Experimental vertical ionization potential for P2 from Ref. [66]. j Experimental vertical ionization potential for SiO from Ref. [67].
PNOF–EKT and KT results are systematically larger than the experimental values. The behavior of HIPs is quite similar to that for the FIP results, discussed previously. For several molecular orbitals (15), PNOF–EKT values are smaller than KT and greater than the experimental data. In case of six molecular orbitals (for example, orbital p for CO and orbital b1 for H2CO) PNOF–EKT values are greater than KT and the experimental data. The exception is P2, for which the PNOF– EKT sg IPs is decreased. The agreement between PNOF andEKT IPs and experimental values is less precise for inner valence MOs. The
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Calculation of vertical ionization potentials with the Piris natural orbital functional Pavel Leiva, Mario Piris * Institute of Physical and Theoretical Chemistry, Friedrich-Alexander-University Erlangen-Nuremberg, Egerlandstrasse 3, 91058 Erlangen, Germany Received 10 January 2006; received in revised form 1 March 2006; accepted 2 May 2006 Available online 10 May 2006
Abstract The Piris natural orbital functional (PNOF) based on a new approach for the two-electron cummulant has been used to predict vertical ionization potentials of 15 molecules. The ionization energies have been calculated using the extended Koopmans’ theorem. The calculated PNOF values are in good agreement with the corresponding experimental data. q 2006 Elsevier B.V. All rights reserved. Keywords: Ionization potentials; Extended Koopman’s theorem; Natural orbital functional
1. Introduction The theoretical investigation in quantum chemistry of natural orbital functional theory (NOFT) [1–10] has increased quickly in the last several years [11–42]. In this formalism, the energy functional of the one-particle reduced density matrix (1-RDM) is expressed in terms of the natural orbitals and their occupation numbers. A great advantage is that the kinetic energy is explicitly defined and the NOFT therefore does not have to invoke the concept of a fictitious noninteracting particle system. Moreover, NOFT incorporates fractional occupation numbers in a natural way, which provides a correct description of both dynamical and nondynamical correlation. Since, the electronic energy is given exactly in terms of the spinless 1-RDM G and the two-particle charge density r2, one can attain a NOFT using a reconstruction functional r2[G]. Such reconstruction functional in terms of two symmetric matrices, D and L, has been recently proposed by Piris [40]. The suggested form of these matrices (as functions of the natural occupation numbers) produces a natural orbital functional (NOF) that reduces to the exact expression for the total energy in two-electron systems [16,43]. One can generalize it to the N-electron systems, except for the off-diagonal elements of D. Alternatively, the mean value theorem and the * Corresponding author. Tel.: C49 9131 8527347; fax: C49 9131 8527736. E-mail address: [email protected] (M. Piris).
0166-1280/$ - see front matter q 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2006.05.001
partial sum rule for matrix D provide a prescription for deriving a practical NOF. The comparison with other theoretical methods showed that Piris NOF (PNOF) provides total energies and dipole moments close to accurate ab initio methods like CCSD(T). Prediction of ionization potentials (IPs) has been a frequent area of application in quantum chemistry [44–48]. Theoretical chemistry provides several alternatives to the calculation of IPs. First, IPs can be computed at any level of theory as the energy differences between the N and (NK1)-particle systems (DE). The energy-difference calculations can be avoided using the electron propagator theory [49], the equation-of-motion method [50], the one-particle Green’s function method [51] or the extended Koopmans’ theorem (EKT) [52,53]. The formalism of EKT not only describes the electron detachment in terms of one-electron quantities, but it also offers the advantage of being highly efficient from the computational point of view. IPs of a number of molecules have been computed from EKT formalism with different approximate electron correlation treatments. Comparison between EKT IP’s and DE IP’s indicate that EKT is exact for the first IP and capable of yielding highly accurate higher IPs [54,55]. The aim of this paper is to apply PNOF to the determination of IPs. This is not the first NOF study of ionization potentials via EKT in molecular systems. Pernal et al. [38] have recently determined the IPs of four selected molecules using NOFs proposed in Refs. [21,28,37]. The present study complements their work: it extends the range of systems to 15 molecules and also it considers the PNOF for the prediction of this property. The performance of the PNOF is established by carrying out a
46
P. Leiva, M. Piris / Journal of Molecular Structure: THEOCHEM 770 (2006) 45–49
statistical analysis of the IP relative absolute errors with respect to the experiment values.
sum rule for off-diagonal elements of matrix D (see Eq. (30) in Ref. [40]) was used. Inserting this expression into Eq. (7), one obtains:
2. Theory
EZ
0 X X ð2ni hii C n2i Kii Þ C ð2nj ni Jij KLji Kij Þ ij
i
We briefly describe here the theoretical framework of our approach. A more detailed description of the PNOF can be found in Ref. [40]. The electronic energy functional for N-electron systems is: X X X E Z2 ni hii C 2 ðnj ni KDji ÞJij K Lji Kij (1) i
ij
if i Z j Z 1
(2)
pffiffiffiffiffiffiffiffi Lji Z ½1K2qð0:5Knj Þqð0:5Kni Þ nj ni ;
if i sj
(3)
where q(x) is the unit step function also known as the Heaviside function. Unfortunately, DjiZnjni violates the obtained sum rule for this matrix in the general case of NO2. This implies that the functional form of nondiagonal elements of D remains unknown for N-electron systems. In contrast, one can assume that the diagonal elements of D equal the square of the occupation numbers Dii Z n2i
(4)
and the functional form (2) and (3) of matrix L can be readily generalized to Lii ZKni
if i Z j
(5)
pffiffiffiffiffiffiffiffi Lji Z ½1K2qð0:5Knj Þqð0:5Kni Þ nj ni ;
if i sj
(6)
By taking into account Eqs. (4) and (5), the energy functional (1) can be expressed as EZ
X
ni ð2hii C Kii Þ C 2
0 X
ðnj ni KDji ÞJij K
ij
i
0 X
Lji Kij
(7)
ij
in which the primes indicates that the iZj term is omitted. It is not evident how to approach Dji, for jsi, in terms of the occupation numbers. Due to this fact, the energy term in Eq. (7), which involves Dji is rewritten as 0 X ij
Dji Jij Z
X i
Ji
0 X j
Dji Z
X
Ji ni ð1Kni Þ
(8)
i
where Ji denotes the mean value of the Coulomb interactions Jij for a given orbital i taking over all orbitals jsi. Here, the
X
ni ð1Kni ÞðKii K2Ji Þ
(9)
i
A further simplification is accomplished by setting Ji zKii =2, which produces:
ij
The states {ji(r)} constitute a complete orthonormal set of single-particle real wave functions, where i denotes the orbital, and {ni} are their occupations. hii is the matrix element of the kinetic energy and nuclear attraction terms. JijZhijjiji and KijZ hijjiji are the electronic repulsion integrals. Note that if DjiZ0 and LjiZnjni, then our reconstruction functional yields the Hartree–Fock (HF) case. From the requirement that for any two-electron system (NZ2) the expression (1) should yield the exact energy functional of Ref. [16], one easily deduces that DjiZnjni and L11 ZKn1
C
EZ
0 X X ð2ni hii C n2i Kii Þ C ð2nj ni Jij KLji Kij Þ i
(10)
ij
The NOF (10) turns out to be identical with the selfinteraction-corrected Hartree functional proposed by Goedecker and Umrigar [16] except for the choice of phases given by the sign of Lji. There are also similarities between this practical NOF and the recently proposed BBC functional [37]. Since, both NOFs derive from the exact two-electron case, matrix L given by Eq. (6) agrees with the first repulsive correction (C1) introduced by Gritsenko et al. to restore the positive phase of the cross products between weakly occupied orbitals. The negative phase keeps otherwise in both functionals. However, the functional form of L is different for many other cross products. In the case of BBC functional the natural orbitals are divided into the groups of strongly occupied (occ), weakly occupied (virt) and frontier (frn) orbitals. Moreover, an orbital may be of the virt and frn type at the same time. If isj and (i,j2occ)n(i2occ, j2frn)n(j2occ, i2frn) then LjiZnjni differing from (6). Another important difference is the diagonal term (iZj), which coincides only if i;frn. The functional form (6) for the matrix elements of L between orbitals with occupation numbers larger than 0.5 gives a wrong description for the lowest occupied levels. The occupation numbers for these levels are identically equal to one. In order to ensure that these occupation numbers only are close to unity an additional term was introduced [40] in the functional form of Lji: pffiffiffiffiffiffiffiffi Lji Z ½1K2qð0:5Knj Þqð0:5Kni Þ nj ni C qðnj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (11) K0:5Þqðni K0:5Þ ð1Knj Þð1Kni Þ; if i sj One has to minimize the energy (10) both with respect to {ji} and {ni}. Bounds on the occupation numbers (0%ni%1) are enforced by setting ni Z ðsin gi Þ2 and varying the {gi} without constraints. Introducing the Lagrange multiplier m with the normalization condition, and the set of symmetric multipliers {3ji} with orthonormality constraints on the real orbitals, the functional whose extremum we seek is given by: " # X X 2 U Z EKm 2 ðsin gi Þ KN K2 3ji ½hji jjj iKdji (12) i
ij
P. Leiva, M. Piris / Journal of Molecular Structure: THEOCHEM 770 (2006) 45–49
The Euler equations for the independent variables {ji} and {gi} are, respectively ! X dU ^ i ni K Z 4 Vj jj 3ji Z 0 (13) dji j dU dE Z sinð2gi Þ K2m Z 0 dgi dni where the one-particle operator V^ is X L~ ij K1 1 ^ C ^ ^ Vð1Þ Z hð1Þ 2nj hjjrK jjiK hjjr jji P 12 12 12 ni j
(14)
(15)
In Eq. (15), the P^ 12 operator permutes electrons 1 and 2. Note that the sum includes the jZi term since we have ~ with diagonal elements equal n2i introduced a new matrix L and off-diagonal elements given by L through Eq. (11). In explicitly dependent on the 1-RDM NOFT [1], the variational derivative of the functional E with respect to the occupation numbers is equal to 2Vii. Moreover, the 1-RDM and the matrix of orbital energies, 3, may be simultaneously brought to diagonal form by the same unitary transformation [3]. In this case, Eq. (14) implies that operator V^ has an essentially degenerate eigenvalue spectrum, i.e. all the natural orbital eigenvalues are the same (m), and are equal to minus the vertical IP [10,56]. As a result, the chemical potential is given by the ratio: mZ
ei ni
(16)
Conversely, we have here an implicitly dependent on the 1-RDM NOFT, so the energy functional still depends on the 2-RDM. As it was pointed out by Donnelly [4], there is a fundamental difference between energy functionals based on 1and on 2-RDMs. Our energy functional (10) is not invariant with respect to a unitary transformation of the orbitals hence Eq. (13) for the optimum orbitals is actually the Lo¨wdin’s equation [57]. In general, this equation cannot be reduced to an eigenvalue problem diagonalizing the matrix 3, although by slight manipulation V^ can be transformed into a Hermitian operator with a nondegenerate spectrum of eigenvalues n [4]. Such construction is provided by the extension of Koopmans’ theorem [52,53]. The equation for the extended Koopmans’ theorem for ionization potentials may be derived by expressing the wavefunction of the (NK1)-electron system as a simple linear combination: X jJNK1 i Z Ci a^ i jJN i (17) i
In Eq. (17), a^i is the annihilation operator for an electron in orbital i, jJNi is the wavefunction of the N-electron system, (JNK1i is the wavefunction of the (NK1)-electron system and {Ci} are a set of coefficients to be determined. Optimizing the energy of the state JNK1 with respect to the parameters {Ci} and subtracting the energy of JN, gives the EKT equations as a
47
generalized eigenvalue problem FC Z nGC
(18)
where n are the EKT ionization potentials. In Eq. (18), the metric matrix is G with the occupation numbers {ni} along the diagonal and zeros in off-diagonal elements, and the transition matrix elements are given by D E ^ a^i jJN ZKVji ni ZK3ji Fji Z JN ja^ †j ½H; (19) This equation can be transformed by canonical orthonormalization using GK1/2. With this transformation Eq. (18) can be written as: F 0 C 0 Z nC 0
(20)
It is now clear from Eq. (19) that the diagonalization of the matrix n with the elements eji (21) nji ZKpffiffiffiffiffiffiffiffi nj ni yields ionization potentials as eigenvalues [4,38]. 3. Results and discussion The simplest treatment for determining vertical IPs is based on Koopmans’ theorem (KT), which states that the IP is given by the HF orbital energy with opposite sign (K3i), calculated in the neutral system [58]. This approach ignores the relaxation of the molecular orbitals after the ionizations. However, it has long recognized that in general there is an excellent agreement between the KT values and the experimental ones because of the fortuitous cancellation of the correlation and relaxation effects. Table 1 lists the obtained vertical IPs by PNOF–EKT together with KT IPs and experimental values. The IPs were calculated using the contracted Gaussian basis set 6-31G** [59], at the near-experimental geometries of neutral molecules given in Ref. [60]. A survey of this table reveals that KT, due to the neglect of orbital relaxation in cationic states, consistently overestimates the first IPs (FIPs), except for C2H2, HCN, Li2 and P2. The prevailing trend is that the values decrease in moving from KT to PNOF–EKT, and then from PNOF– EKT to experimental data. Conversely, in case of C2H2, Li2 and P2, the FIPs increase in moving from KT to PNOF– EKT, and then from PNOF–EKT to experimental values. For CH4 and HCN molecules, the PNOF–EKT IPs are greater than the KT and experimental results, whereas in case of HCl and SiO the PNOF–EKT FIPs are smaller than the other values. Generally, the PNOF–EKT FIPs move closer to experimental data. In fact, the relative error of the PNOF–EKT values is smaller than the relative error of KT ones. The KT and PNOF–EKT methods show relative absolute difference with respect to the experiment of 5.6 and 3.5%, respectively. Table 1 lists also the higher IPs (HIPs) calculated via KT and PNOF–EKT methods for the 15 molecules. In general, the
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P. Leiva, M. Piris / Journal of Molecular Structure: THEOCHEM 770 (2006) 45–49
Table 1 Vertical ionization potential in electron volt Molecule
MO
Koopmansa
PNOF– EKTb
Expc
C2H2
pu sg su t2 a1 s p s pg pu sg p s b2 b1 a1 b2 b1 a1 b2 p s s p s s s s sg pu su a1 e pu sg s p
10.86 18.27 20.73 14.75 25.58 14.90 17.22 21.67 17.88 21.82 20.19 17.06 20.23 11.87 14.46 17.58 18.72 13.53 15.56 19.10 12.93 17.00 30.42 13.32 15.63 22.01 4.84 8.16 17.13 16.63 21.11 11.44 16.87 10.14 11.14 11.77 12.76
11.12 18.24 20.12 14.86 25.14 14.59 17.28 21.58 16.74 21.22 22.21 16.82 19.99 11.54 14.63 17.44 18.40 13.08 15.54 18.93 12.51 16.73 26.57 13.66 15.67 21.64 4.90 8.00 16.74 17.14 20.96 11.05 16.93 10.39 10.63 11.41 12.21
11.49 16.70 18.70 14.40 23.00 14.01 16.85 19.78 15.87d 18.80d 21.10d 16.19 19.90 10.90 14.50 16.10 17.00 12.78 14.83 18.72 12.77e 16.60e 25.80e 13.61f 14.01f 19.86f 5.14g 7.70h 15.60 16.68 18.78 10.80 16.80 10.65i 10.84i 11.61j 12.19j
CH4 CO
F2
FH H2CO
H2O
HCl
HCN HCN Li2 LiH N2
NH3 P2 SiO
relative errors are slightly higher for HIPs (5.5%) than the corresponding for FIPs (3.5%). We find also this trend for the KT results (7.0% for HIPs and 5.6% for FIPs). Important cases are the F2 and N2 molecules. It is wellknown that the KT sg and pu IPs are in the wrong order for both molecules (see Table 1). PNOF–EKT calculations on F2 and N2 give valence shell IPs, which are in the correct order, and in general a numerical improvement is obtained over KT IPs. In case of the orbital pu for N2, the KT IPs is closer to experimental data but due to the mentioned wrong ordering. The PNOF–EKT and KT values (including all IPs) show relative errors with respect to the experiment of 4.7 and 6.5%, respectively. In general, the results are in good agreement with the corresponding experimental vertical IPs considering the small basis sets used for these calculations. 4. Conclusions The recent obtained PNOF [40] was used to determine vertical ionization potentials for a set of 15 selected molecules at the experimental geometries. The IPs obtained from PNOF–EKT method are compared with the experimental IPs. The agreement between our PNOF– EKT IPs and those obtained experimentally is somewhat less for higher IPs than for the first IPs. The PNOF–EKT method under study provide an improvement in the IPs over the KT approach. In general, we observed that the overall trends with the introduction of the electronic correlation by PNOF are satisfactory. The reliability of the PNOF in the prediction of the studied property has been illustrated. It is confirmed that EKT calculations produce IPs of comparable accuracy but at a substantially lower computational cost. Acknowledgements
a
K3HF i . b Vertical ionization potentials obtained from the extended Koopmans’ theorem. c Experimental vertical ionization potentials from Ref. [54]. d Experimental vertical ionization potential for F2 from Ref. [61]. e Experimental vertical ionization potential for HCl from Ref. [62]. f Experimental vertical ionization potential for HCN from Ref. [63]. g Experimental vertical ionization potential for Li2 from Ref. [64]. h Experimental adiabatic ionization potential for LiH from Ref. [65]. i Experimental vertical ionization potential for P2 from Ref. [66]. j Experimental vertical ionization potential for SiO from Ref. [67].
PNOF–EKT and KT results are systematically larger than the experimental values. The behavior of HIPs is quite similar to that for the FIP results, discussed previously. For several molecular orbitals (15), PNOF–EKT values are smaller than KT and greater than the experimental data. In case of six molecular orbitals (for example, orbital p for CO and orbital b1 for H2CO) PNOF–EKT values are greater than KT and the experimental data. The exception is P2, for which the PNOF– EKT sg IPs is decreased. The agreement between PNOF andEKT IPs and experimental values is less precise for inner valence MOs. The
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