Insights into Direct Methods for Predictions of Ionization Potential and

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Spectroscopy and Photochemistry; General Theory

Insights into Direct Methods for Predictions of Ionization Potential and Electron Affinity in Density Functional Theory Neil Qiang Su, and Xin Xu J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.9b01052 • Publication Date (Web): 06 May 2019 Downloaded from http://pubs.acs.org on May 7, 2019

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The Journal of Physical Chemistry Letters

Insights into Direct Methods for Predictions of Ionization Potential and Electron Affinity in Density Functional Theory Neil Qiang Su† and Xin Xu*,† † Collaborative Innovation Center of Chemistry for Energy Materials, Shanghai Key Laboratory of Molecular Catalysis and Innovative Materials, Ministry of Education Key Laboratory of Computational Physical Sciences, Department of Chemistry, Fudan University, Shanghai, 200433, China E-mail: [email protected]

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Abstract Vertical ionization potential (IP) and electron affinity (EA) are fundamental molecular properties, while the ∆ method and the direct method are the widely used approaches to compute these properties. The ∆ method is calculated by taking the total energy difference of the initial and final states, whose reliability is seriously affected by the issue associated with the imbalanced treatment of these two states. The direct method based on the derivatives involving only one single state calculation can yield a quasi-particle spectrum, whose accuracy, one the other hand, is mostly affected by the levels of approximate molecular structure theories. Due to the aforementioned issues, EA prediction can be particularly problematic. Here we present, for the first time, the analytic theory on the derivation and realization of generalized Kohn-Sham (KS) eigenvalues of doubly hybrid (DH) functionals that depend on both occupied and unoccupied orbitals. The method based on the KS eigenvalues of neutral systems, coined as the NKS method, is found to suffer little from the imbalance issue, while it is only the NKS method that can offer accurate EA prediction from a good functional approximation, such as the XYG3 type of DH functionals. Being less sensitive to the size of basis sets, the NKS method is of great significance for its application to large systems. The insights gained in this work are useful for the calculation of properties associated with small energy differences, while emphasizing the importance of the development of generalized functionals that rely on both occupied and unoccupied orbitals. TOC GRAPHICS


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Keywords Kohn-Sham method, the derivative approach, doubly hybrid functional, the Janak’s theorem, eigenvalue, orbital energy


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Ionization potential (IP) and electron affinity (EA) are fundamental electronic properties1-4 that are intimately related to a myriad of crucial phenomena in chemistry, physics, and materials science, such as energy and charge transfer, ionizing radiation, photocatalysis, photoluminescence, etc. In practice, theoretical calculations, along with experiments, play the critical role to achieve in-depth insights into these phenomena, thus benefitting the development of science and technology, which, in turn, benefits the human life. Hence, numerically accurate yet computationally efficient prediction of IPs and EAs for a wide range of systems, from atoms to molecules and solids, is of great importance, which, however, poses a significant challenge to the electronic structure theory. Ab initio methods based on wave-functional theory, such as the equation-of-motion coupled cluster (EOM-CC) methods5,6, can offer excellent accuracy for IP and EA predictions, but the computational complexity severely limits the scale of the system under study. Meanwhile, density functional theory (DFT) in the Kohn-Sham (KS) or generalized KS (GKS) framework1,7-9, due to the favorable balance between accuracy and efficiency, has become the method of choice in the many-electron problem. In the practical application of (G)KS-DFT, the exchange-correlation (XC) energy must be obtained by a density functional approximation (DFA) whose quality directly affects the prediction accuracy for various properties. For IP and EA calculations, in addition to different choices of DFAs, different calculation approaches that will be introduced below can lead to distinct prediction accuracies. The Δ method is a straight approach to calculate IPs as well as EAs, via the energy difference between two calculations of the initial and final states. This method has been extensively applied in the literature, such as Refs. 10-19. One of the disadvantages of the Δ method is that the energy difference is typically much smaller than the total energy of either the initial state or the final state,


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thereby any imbalanced treatment of the two states can result in a significant impact on the calculation result. For example, for EA calculations, the same basis set, in use for both neutral and anionic species, tends to be more deficient for the anions20-24, resulting in the underestimation of the anion stabilities and thus the underestimation of the EA values. Typically, the Δ method often fails to offer a reliable prediction for an EA of near-zero positive value, even making the additional electron unbound. Aiming to avoid the imbalance issue of the Δ method, direct methods, where IPs and EAs are computed without explicitly involving the energy difference of the initial and final states, have been applied. The integration approach25-28 calculates the energy difference through numerical integration of the energy derivative with respect to a chosen parameter that defines a path connecting the initial and final states. For IP and EA calculations, an obvious choice for the parameter is the occupations of occupied and unoccupied orbitals, respectively, thus the integration path between two end points is then characterized by fractional charges. This approach, first used at the Hartree-Fock (HF) level without correlation25, has been extended to the Møller-Plesset theory with dynamic correlation up to second order (MP2)29 for better IP and EA prediction26-28. The combination of this approach with DFT methods will be further discussed in this work. A more popular direct method for the IP and EA prediction is through (G)KS eigenvalues, which are the energy gradients with respect to orbital occupations30-32. It has a long tradition to construct an approximate quasi-particle spectrum, i.e., a spectrum containing the negative of the IPs and EAs, from the (G)KS eigenvalue spectrum associated with the neutral molecule, as exemplified by Refs. 33-42. Based on the Perdew-Parr-Levy-Balduz (PPLB) condition43-46 that requires the total energy to be piecewise straight lines interpolating between adjacent integer points, the chemical potentials are related to the predicted IP and EA, namely 𝜇 ― = -IP and 𝜇 + = -EA43.


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Many efforts have been devoted towards the physical content of eigenvalues within KS framework47-53. The further connection between the chemical potentials and eigenvalues of HOMO (i.e., the highest occupied molecular orbital) and LUMO (i.e., the lowest unoccupied molecular orbital) for both KS and GKS frameworks was constructed in Ref. 32. It proves that the (G)KS HOMO eigenvalue is equal to the chemical potential for electron removal, namely 𝜀(𝐺)𝐾𝑆 𝐻𝑂𝑀𝑂 = (∂𝐸/∂𝑁)𝑉― = 𝜇 ― ; and the (G)KS LUMO eigenvalue is equal to the chemical potential for + + electron addition, namely 𝜀(𝐺)𝐾𝑆 𝐿𝑈𝑀𝑂 = (∂𝐸/∂𝑁)𝑉 = 𝜇 . However, such connection cannot be

strictly extended to other eigenvalues. In practical applications, commonly used functionals, such as local density approximations (LDAs), generalized gradient approximations (GGAs) and hybrid functionals, tend to underestimate IPs and overestimate EAs. This failure has been attributed to the delocalization error54 inherent in the approximate functionals, which manifests itself as a convex deviation from the PPLB condition (See Fig. S1 in the Supporting Information (SI) for the fractional charge behaviors of some functionals studied in this work). In comparison to HOMO (LUMO), the overestimation (underestimation) of other eigenvalues in the hole (particle) part becomes more serious when serving as approximations to the corresponding IPs (EAs). To achieve better approximations to principal IPs (or EAs) with commonly used functionals, the Slater’s transition state method55-57, which consists of solving the (G)KS equations with half an electron removed from the corresponding occupied orbital (or added to the corresponding unoccupied orbital), is an effective approach to provide "better" eigenvalues. For simplicity, below we will refer to the (G)KS eigenvalues of the N-electron systems (the neutral systems) as NKS, while eigenvalues computed by the Slater’s transition state method as STS. Corresponding to the above integration approach, both NKS and STS are classified as the derivative approach.


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A distinct advantage of NKS is that the entire (G)KS eigenvalue spectrum, serving as approximations to IPs and EAs for different states, can be obtained with only one calculation for the ground state of the neutral system. In contrast, the Δ method, as well as other direct methods, i.e. the integration and STS methods, needs another calculation at a specific choice of orbital occupations for each IP or EA prediction. Especially, non-Aufbau choices of occupations are required for IPs below HOMO and EAs above LUMO, which would cause serious convergence problem as high-lying solutions have a very strong tendency to collapse to lower-lying states. Therefore, NKS has the significant computational advantage over other methods. However, this ease of computation is usually inversely related to the accuracy in the application of DFT with common approximations. The performance of commonly used DFAs as exemplified by B3LYP58-61 on the first EA of C atom is shown in Fig. 1. The Δ method underestimates the experimental value, seriously affected by the imbalance issue (See Fig. 2 for more results and discussion). For the integration approach, the result is getting closer to that of the Δ method as the number of grid points used for numerical integration increases. NKS seriously overestimates the experimental value due to the delocalization error in B3LYP. In contrast, STS can effectively avoid the impact of the delocalization error on the EA calculation, giving the similar result as the Δ method. The fact that the integration and STS methods show similar performance here as the Δ method indicates that they may have a similar effect of the imbalance issue as the Δ method. To better understand these direct methods, insights from better DFAs with less delocalization error are essential.


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Figure 1. Comparison between the integration and derivative approaches on predicting EA of C atom. (a) The integration approach with increasing number of equidistant grids is checked; (b) The derivative approach, the negative of the LUMO eigenvalue, with increasing occupation is checked. The results of the ∆ method, as well as the experimental value are provided for comparison. All energies are in eV. The basis set used is the same as Refs. 26-28. See Supporting Information (SI) for the details.

Advances in constructing better functionals have led to great improvement on predicting IPs and EAs with eigenvalues62-71. Long-range corrected (LRC) hybrid functionals72-76, especially tuned LRC hybrid functionals that employ a range-separation parameter tuned separately for each system of interest, have been demonstrated to yield accurate ionization potentials and fundamental gaps for a wide range of finite systems69,70,77,78. However, orbitals of different characters and symmetries require significantly different range-separation parameters and fractions of the exact exchange, thus tuned hybrid functionals cannot offer a balance accuracy for the entire quasi-


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particle spectrum79. Recent work on evaluating the fractional charge behavior of DFT methods68 shows that doubly hybrid (DH) functionals80-82 significantly improve results from commonly used (semi-)local and hybrid functionals on describing systems with fractional charges; especially, the XYG3 type of DH (xDH) functionals82-87 can mostly restore the linear behavior of the PPLB condition. It thus implies that the xDH functionals are able to provide reliable HOMO and LUMO eigenvalues for the first IP and EA prediction. Therefore, the theory on the derivation and realization of DH GKS eigenvalues, which allows the prediction of the whole quasi-particle spectrum, should be interesting and will be presented here. This work should be of great importance for the development and applications of functionals that depend on both occupied and unoccupied orbitals. The focus of this work is twofold. On the one hand, we derive and implement the DH “eigenvalues”, and evaluate the performance of the DH functionals on IP and EA predictions. On the other hand, this article seeks to gain a deeper understanding of the Δ and direct methods, and their connection to approximated functionals in use. Insights gained from this work should be useful for both the development and application of DFAs. Next, the DH eigenvalues will be derived from the energy gradients with respect to orbital occupations, which generalizes the eigenvalue calculation for functionals of both occupied and unoccupied orbitals. Detailed derivation can be found in Sec. I in the SI. Analogous to commonly used functionals, we call the gradients as orbital energies or GKS eigenvalues. The exchange-correlation (XC) energy of the DH functionals takes the form:

ExcDH  Excn,Hyb  EcPT 2  cxn E xHF  E xcn,DFA  cc EcMP 2 ,

(1)


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and the total energy is evaluated non-variationally by inserting the orbitals and eigenvalues from another functional used for the self-consistent filed (SCF) calculation,

Excs,Hyb  cxs ExHF  Excs,DFA .

(2)

Here, cx and cc are the mixing coefficients for the HF-like exchange ExHF and the MP2-like correlation EcMP 2 , respectively; ExcDFA stands for a (semi-)local DFA; The superscript s or n emphasizes that it is for the SCF calculation to generate orbitals or for the non-variational energy evaluation. Note that if Excs , Hyb  Excn , Hyb holds, this is a B2PLYP81 type of DH functionals; otherwise, this is an xDH82 functional. Unlike the common DFT calculation yielding the eigenvalues directly from the SCF calculation, the eigenvalues  ts , Hyb result from the SCF functional of Eq. 2 should be further updated to the desired DH eigenvalues by taking into account the additional contribution from the PT2 (the perturbation theory to 2-nd order) part and the inconsistency between Excs , Hyb and

Excn,Hyb . The final DH eigenvalues take the following form,

 tDH 

dEtotDH   ts,Hyb   td ,DH   tid ,DH , dnt

(3)

where the correction includes the direct and the indirect contributions. The direct correction comes from the explicit dependence on orbital occupations,

 td ,DH   td ,Hyb   td ,PT 2 ,

(4)

with  td ,Hyb from the inconsistency between Excs , Hyb and Excn,Hyb ,









 td ,Hyb  t cxn  cxs vxHF  vxcn,DFA  vxcs,DFA t ,

(5)


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and  td ,PT 2 from the PT2 part, n j 1 na  1 nb  tj ab ni n j 1 nb  ij tb c c  td ,PT 2  c  s,Hyb s,Hyb s,Hyb s,Hyb  c  s,Hyb s,Hyb s,Hyb s,Hyb . 2 jab  t   j   a   b 2 ijb  i   j   t  b 2

2

(6)

Here, vxHF is the non-local HF exchange potential, vxcDFA is the semi-local DFA potential, and

pq rs  pq rs  pq sr

is the antisymmetrized two-electron integral. Similar results for

 td ,PT 2 in the MP2 theory has been reported several times before in the literature26-28,88. The indirect correction comes from the indirect dependence on occupations, through both orbitals

and eigenvalues

,

,

(7)

where coupled-perturbed KS (CP-KS) equations89-93 are required to take into account the orbital relax due to the change of occupation dnt . Formally, each DH eigenvalue needs to solve a set of CP-KS equations corresponding to a specific orbital, thus N set of equations are required to complete the whole eigenvalue spectrum. Here, by making use of a modified Z-vector method (see SI for more details), the indirect correction can be eventually formulated as27,28 PT 2 PT 2 PT 2 PT 2  tDH ,id  Po-o .Bo-o,t  Pv-v .B v-v,t  Pv-o .B v-o,t  Po-v .Bo-v,t ,

(8)

where P PT 2 is the difference density matrix, with the occupied-occupied (o-o) and virtual-virtual (v-v) blocks defined, respectively, by27,28


11


 cc   ni nl 1 na 1 nb tilabtilab ,   2 lab PijPT 2     cc  ni n j nl 1 na 1 nb tilabt ab jl ,   2 lab













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for i  j

,

(9)

for i  j

and  cc  2 PabPT 2 =   cc  2

 n n 1  n 1  n  t i

j

a

d

for a  b

ad ad ij ij

t ,

ijd

 n n 1  n 1  n 1  n  t i

j

a

b

d

, ad bd ij ij

t ,

(10)

for a  b

ijd

ab with tij being the double substitution amplitude



tijab  ij ab

s,Hyb i



  s,Hyb   as,Hyb   bs,Hyb . j

(11)

PT 2 PT 2 The symmetry requirement makes Po-v = transpose( Pv-o ). And the v-o block of the difference PT 2 density matrix, Pv-o , can be obtained by solving the following equations,

PT 2 Pv-o A  LPT 2 ,

(12)

where LPT 2 is the total Lagrangian combining the contribution from the PT2 part and the inconsistency between Excs , Hyb and Excn,Hyb ,

2 LPT ai  







cc  ni n j nl 1 na 1 nb t abjl ib jl 2 jlb

     n 1 n  P  aj il  c aj li  G    n 1 n  P  ab id  c ab di  G   F





cc ni n j 1 na 1 nb 1 nd tijcb db aj  2 jbd i

a

( 2) jl

i

a

( 2) bd

s x

.

(13)

s jl ,ai

jl

s x

s bd ,ai

n ai

bc


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A of Eq. 12 and Bt of Eq. 8 are defined, respectively, by





s  , Apq,rs    rs,Hyb   ss,Hyb  rp sq  ns 1 nr   2 ps qr  cxs ps rq  cxs pr sq  2G pq,rs

(14)

and s Bpq,t  pt pt  cxs pt tp  G pq,tt .

(15)

Therefore, only one, instead of N, set of equations need to be solved for P PT 2 that is independent of the change of occupation dnt , and then the correction to all eigenvalues can be evaluated by inserting P PT 2 into Eq. 8. In the above equations, G s is a response-type matrix obtained from derivative of the Fock matrix associated with the SCF functional Excs , Hyb 92-94, while F n is a Focktype matrix derived from the hybrid part Excn,Hyb of DH functionals. G s is specific to the DH functional, which is absent in the MP2 method using the HF orbitals, while F n and  td ,Hyb are specific to the xDH functional, which vanish in the B2PLYP functional. The performance of DH functionals (e.g. B2PLYP and XYG3) on the EA of C atom can be found in Fig. 1, along with the results from MP2. Affected by the basis set imbalance between the neutral and anionic species, XYG3 also underestimates the experimental value with the Δ method. The integration approach shows similar effect on the imbalance issue, the result becomes close to the Δ method when enough grid points are used. Similar results can be observed for all the other functionals tested in Fig. 1, indicating that the imbalance issue in the ∆ and integration methods is ubiquitous, no matter which functional approximation, or even the exact functional, is used. For the derivative approach, the XYG3 eigenvalue when LUMO is completely empty, i.e. NKS, provides an excellent approximation to the experimental EA. This good performance indicates that


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there exists much less delocalization error in the XYG3 than B3LYP (See Fig. S1 in SI).68 However, as the LUMO occupation increases, the results become worse, and a result similar to the ∆ method can be obtained when the occupation is 0.5 (i.e. STS). This infers that the derivative approach with nonzero LUMO occupation still suffers from the imbalance issue as the ∆ method, which becomes more serious as the LUMO occupation increases. To further understand the imbalance issue of the derivative approaches, Fig. 2 compares the basis set dependence of NKS and STS. As can be seen, NKS shows less basis set dependence, and the predicted EAs change only a little as the basis set changes. The nearly unchanged values predicted with the additional augmented functions in the basis sets imply that NKS is insensitive to the number of diffuse functions. In contrast, STS are more affected by the basis sets. By observing, we find that the basis set dependence and the predicted EAs by STS are very similar to the ∆ method, which again indicates that STS cannot solve the imbalance issue. Although the ∆ method suffers more seriously from the delocalization error for larger systems54,67, both ∆ method and STS methods can effectively avoid the impact of the delocalization error on the EA prediction for small molecules tested here. Hence, they require larger basis sets to achieve good prediction accuracy due to the imbalance issue. NKS suffers less from basis set driven error, however, the prediction accuracy is seriously affected by the DFA driven error. Therefore, only a good functional approximation, such as XYG3, can provide an accurate EA prediction by NKS, without requiring a very large basis set, which is of great significance for the application to large systems.


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Figure 2. Basis set dependence of the derivative approaches on predicting the EA of C atom. NKS predicts the EA by the negative of the eigenvalue of empty LUMO, while STS by the negative of eigenvalues of half-occupied LUMO. The ∆ method results are also provided for comparison. Correlation-consistent basis sets95,96 are used (See SI for more details). The lines connecting adjacent points are just guides for the eyes. All energies are in eV.

Here we also conducted more tests on the first IPs and EAs, fundamental gaps, and core and valence IPs. Table 1 summarizes the mean absolute errors (MAEs) of some functionals against a set of 13 species consisting of atoms, molecules and radicals. More details and more test results can be found in Sec. II of SI. Since the integration approach gives the same results as the ∆ method, only the ∆ , NKS and STS methods are discussed here. As shown in Table 1, the statistics for the Δ and STS methods are generally similar. At a large basis set such as VQZ95,96, the Δ and STS methods are actually quite satisfactory and different levels of electronic theory lead to similar


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accuracy. In fact, the Δ and STS methods are especially effective in predicting the first IPs and the CVIPs that are less affected by the imbalance issue and have large values in magnitude. Table 1. Mean absolute errors (MAEs) of tests on the first IPs, the first EAs, fundamental gaps (GAPs), and core and valence IPs (CVIP). a

∆b

STSc

NKSd

IP

EA

GAP

CVIP

MP2

0.20

0.47

0.38

0.77

B3LYP

0.35

0.34

0.67

0.65

B2PLYP

0.20

0.47

0.58

0.55

XYG3

0.17

0.46

0.55

0.54

MP2

0.24

0.32

0.32

1.11

B3LYP

0.34

0.36

0.67

0.78

B2PLYP

0.21

0.42

0.57

0.62

XYG3

0.20

0.40

0.54

0.55

MP2

0.84

0.58

0.86

3.56

B3LYP

3.70

2.77

6.42

8.18

B2PLYP

2.35

1.37

3.72

5.52

XYG3

0.78

0.11

0.79

1.89

a. Detailed calculation results can be found in Tables S1-S5 of SI for IP, Tables S6-S10 of SI for EA, Tables S11-S15 of SI for GAP, and Tables S16-S20 of SI for CVIP. Reference values are from Refs. 97-103. All the data are in eV. The basis set used is the same as Refs. 26-28. b. The ∆ method calculates all these quantities as energy differences: IP by 𝐸0(𝑁 ― 1) ― 𝐸0(𝑁), EA by 𝐸0(𝑁) ― 𝐸0(𝑁 + 1), GAP by 𝐸0(𝑁 ― 1) + 𝐸0(𝑁 + 1) ― 2𝐸0(𝑁), and CVIP by 𝐸𝑘 (𝑁 ― 1) ― 𝐸0(𝑁). c. STS calculates all these quantities from eigenvalues of the KS equations with half electron removed from the corresponding occupied orbital (or added to the corresponding unoccupied orbital). d. NKS calculates all these quantities from KS eigenvalues of the neutral systems: IP by minus HOMO eigenvalue, EA by minus LUMO eigenvalue, GAP by LUMO-HOMO gap, and CVIP by minus occupied orbital eigenvalues.


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On the other hand, we are particularly interested in the NKS method, as this method can yield the whole spectrum upon a single state calculation. It is important to notice that NKS is greatly affected by the DFA used. Hence, XYG3 outperforms B2PLYP which, in turn, outperforms B3LYP (see Table 1). Especially for EAs, only NKS with XYG3 can provide reliable predictions (the MAE is only 0.11 eV, much smaller than 0.40 eV by STS with XYG3), which benefits from the small delocalization error of XYG3 and the minor imbalance issue of NKS. In comparison, the considerable delocalization error in B2PLYP and B3LYP makes them impossible to achieve reliable IP and EA predictions through NKS. Data in Table 1 reflect the great importance for further functional improvement to achieve even better NKS predictions. The functional dependence is once again illustrated in Fig. 3(a) with NKS predicted fundamental gaps, where improvement is clearly seen from B3LYP to B2PLYP and to XYG3. Fig. 3(b) depicts NKS prediction for valence IPs of n-butane. The excellent performance with XYG3 demonstrated that NKS, when combined with a good functional approximation, will have advantages in both computational efficiency and computational accuracy.


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Figure 3. (a) Calculated LUMO-HOMO gaps versus experimental fundamental gaps. The x axis stands for experimental fundamental gaps while the y axis for the predicted results. Detailed test results can be found in Tabs. S11-S15 of SI. (b) Valence IPs predicted by NKS, i.e. the negative of KS eigenvalue spectrum of the neutral systems, in comparison with experimental results. Experimental results are from Refs. 97-101 for fundamental gaps, and from Ref. 104 for n-butane. All data are in eV.

In summary, we presented the theory on the derivation and realization of DH eigenvalues, with which, far-reaching insights into both the integration and derivative approaches for the IP and EA prediction have been achieved. The integration method converges to the ∆ method when using enough grid points, so this method cannot solve the imbalance issue associated with the imbalanced treatment of the final and initial states in the ∆ method. The Slater’s transition state method can effectively avoid the impact of the notorious delocalization errors on IP and EA prediction for molecules tested here, however, it requires a large basis set to achieve good prediction accuracy due to the imbalance issue. Conversely, NKS, i.e. the KS eigenvalues of the neutral systems, suffers much less from the imbalance issue, however, the prediction accuracy is seriously affected by the DFA used. Among all the test methods, only good functional approximations, such as the xDH functionals, can provide accurate EA prediction by NKS, without requiring a very large basis set, which is of great significance for large system calculations. The insights gained in this work are useful for the calculation of properties associated with small energy differences, while emphasizing the importance of the development of generalized functionals that rely on both occupied and unoccupied orbitals.

Acknowledgement


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We appreciate the supported from National Natural Science Foundation of China (Grant 21688102), the Science Challenge Project (Grant TZ2018004) and the National Key Research and Development Program of China (Grant 2018YFA0208600).

Supporting Information Available SI.pdf: Detailed derivation of DH eigenvalues and more test results.

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Insights into Direct Methods for Predictions of Ionization Potential and

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