Excited State Vibrational Frequencies: Restricted Virtual Space Time

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A: New Tools and Methods in Experiment and Theory

Excited State Vibrational Frequencies: Restricted Virtual Space Time-Dependent Density Functional Theory Magnus W.D. Hanson-Heine J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b09642 • Publication Date (Web): 13 Mar 2019 Downloaded from http://pubs.acs.org on March 15, 2019

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The Journal of Physical Chemistry

Excited State Vibrational Frequencies: Restricted Virtual Space Time-Dependent Density Functional Theory Magnus W. D. Hanson-Heine* School of Chemistry, University of Nottingham, University Park, Nottingham NG7 2RD. [email protected]

ABSTRACT: Calculating accurate vibrational frequencies for molecules with electronically excited states has an important function in many areas of photochemistry. However, calculations are often limited to smaller molecules due to the rapid growth in the degrees of freedom that must be taken into account to accurately describe larger systems. The applicability of the restricted virtual space (RVS) approximation has been studied within adiabatic linearresponse time-dependent density functional theory (TDDFT) when calculating excited state nuclear vibrational frequencies. Using the S1 and T1 electronic states of CO, CN-, HOF, H2CS, and C2H4 as representative examples, it is found that vibrational frequency calculations are particularly sensitive to this approximation, with no more than 10 to 20 % of orbitals recommended for safe removal without a priori knowledge when using the 6-311+G(d,p) and aug-cc-pVTZ basis sets. Higher frequency vibrations such as those with a high degree of CH bond stretching character are found to be less sensitive to the RVS than the lower frequency vibrations, and several of the triplet states are also found to be less sensitive to this approximation than their equivalent singlet states. Occupied core orbitals and high energy virtual orbitals with core character can also be removed without introducing significant error.

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Introduction Understanding the vibrational frequencies of systems with electronically excited structures plays a vital role in understanding many areas of chemistry. Experimental infrared data allows for excited states and reaction intermediates to be characterized on ultra-fast time scales,1-3 understanding vibrational structure plays an important role in interpreting vibronic coupling within electronic spectroscopy,4-6 and vibrational zero-point energy is also important for calculating thermochemical properties,7-8 to name a few of the applications. Wavefunction methods for calculating excited states, such as complete active space selfconsistent-field (CASSCF), coupled with multiconfigurational perturbation theory (CASPT2),9 and multireference configuration interaction (MRCI)10, are accurate, but can become prohibitively expensive as the size of the system increases. This is particularly true for calculating properties that require extensive sampling of the nuclear potential energy surface (PES) within the Born-Oppenheimer approximation, of which vibrational frequencies are a prime example. Methods based on density functional theory (DFT) generally provide a reasonable compromise between accuracy and computational cost which is better suited to calculating vibrational frequencies, and the majority of excited state frequency calculations make use of adiabatic linear-response time-dependent DFT (TDDFT).11 Cheaper methods such as those based on excited state Kohn-Sham DFT (eDFT/ΔSCF) are known to perform well for describing vibrational properties of states that are essentially well described by a single reference set of orbitals.12-13 eDFT has also recently shown promise in outperforming TDDFT for the calculation of certain S1 state vibrational frequencies,14-15 however, this method is still limited by the use of a single Slater determinant as well as being known to suffer from variational collapse.16 Page | 2 ACS Paragon Plus Environment

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In order to extend TDDFT calculations to larger systems and to allow for more extensive properties to be calculated, several approximations have been developed to reduce the computational cost without significantly sacrificing the accuracy of the calculation. Among these one of the more commonly used approximations is the Tamm-Dancoff approximation (TDA),17 which reduces the dimension of the TDDFT equations by a factors of 2, while avoiding the triplet state instabilities found with full TDDFT.18 The effects of the TDA have been benchmarked for calculating excited state frequencies, and have been found to give a more consistent level of accuracy when describing the excited state vibrational frequencies of a series of different S1 states compared to TDDFT.19 Jacquemin and co-workers also compared vibrationally resolved absorption and emission spectra of several prototypical conjugated molecules using both TDDFT and the TDA, and found that the spectral features unrelated to energy remained the same.20 For these reasons the TDA has been used in this study. A second commonly used approximation is the restricted virtual space approximation (RVS), or restricted excitation space approximation (RES), in which a reduced excitation subspace of orbitals is included within the TDDFT calculation.21 The RVS method has been applied for calculating the excitation energies of diverse systems including solvated molecules,22-23 molecules on surfaces,24-26 protein environments,27-28 and core excited states.29-30 Recently the RVS has also been used in conjunction with integral screening and integration grid augmentations to produce further reductions in computational time for larger systems.31 Robinson examined the effects of reducing the excitation space during the evaluation of the nuclear gradient when calculating excited state geometries and emission energies using the TDA, and found a maximum deviation from all-orbital TDDFT/TDA of less than 0.01 Å in bond-lengths and less than 0.5° in bond angles, with efficiency gains of 15-30% for most of the systems studied.32 However, to the knowledge of the author no benchmarking study has yet been done testing the efficacy of the RVS for the calculation of vibrational data, which is the Page | 3 ACS Paragon Plus Environment


focus of this work. While it is tempting to try and relate the interactions of specific orbitals to bonding effects causing errors within specific vibrational modes, the chemically unintuitive and delocalized nature of the high energy canonical virtual orbitals produced by non-minimal basis sets, as well as the increasingly delocalized nature of the normal modes seen for larger molecules, makes relating specific orbital characteristics to specific vibrational modes impractical. Although the possibility remains that localized orbital TDDFT schemes and the various localized normal mode schemes that have recently been developed might be able to shed some light on these relationships,33-40 the scope of this paper is restricted to the general trends observed when using standard canonical orbital TDDFT codes.

Computational Details Geometry optimizations, energy calculations, and vibrational frequency calculations were carried out using a developmental version of the Q-Chem 5.0 quantum chemical software package.41 Unless otherwise stated, these calculations were performed using the B3LYP exchange-correlation functional and 6-311+G(d,p) basis set.42-43 Excited state geometries and harmonic vibrational frequencies were evaluated numerically using finite differences of the TDDFT/TDA excitation energies unless also otherwise stated in the text. Calculations were carried out on the CN-, CO, HOF, C2H4, and H2CS molecules in their respective S1 and T1 states, as they represent a range of different excitation types. Various levels of the RVS approximation were also applied to the calculations, where orbitals in the virtual space were progressively removed from the excitation space. Orbitals were removed starting with the highest energy virtual orbitals in steps of 5 %, rounded down to the nearest whole number unless otherwise stated. The effects of removing the core orbitals from the


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excitation space were also tested. Within TDDFT, the excitation energies and associated transition dipole moments are obtained from the following equation:11

[

𝐀 𝐁∗

][ ]

𝐁 𝐗 1 𝐀∗ 𝐘 = 𝜔 0

[

][

]

0 𝐗 ―1 𝐘

1

where 𝐴𝑖𝑎,𝑗𝑏 = 𝛿𝑖𝑗𝛿𝑎𝑏(𝜖𝑎 ― 𝜖𝑖) + (𝑖𝑎│𝑗𝑏) + (𝑖𝑎|𝑓𝑋𝐶|𝑗𝑏) 𝐵𝑖𝑎,𝑗𝑏 = (𝑖𝑎│𝑏𝑗) + (𝑖𝑎|𝑓𝑋𝐶|𝑏𝑗)

2

where the two-electron integrals are again given in Mulliken notation.11 When the TDA is applied, the B matrix elements are set to zero giving, 3

𝐀𝐗 = 𝜔𝐗.

When the RVS or RES is applied and a subset of the orbital excitation space is chosen, these TDDFT and TDA equations become

[𝐁𝐀

∗

][ ] [

𝐁 𝐗 1 =𝜔 0 𝐀∗ 𝐘

][

]

0 𝐗 ―1 𝐘

4

and 𝐀𝐗 = 𝜔𝐗

5

𝐀 = 𝐴𝑖𝑎,𝑗𝑏 𝐁 = 𝐵𝑖𝑎,𝑗𝑏

6

where

in which the indices 𝑖 and 𝑎 denote subsets of the occupied and virtual orbitals, respectively.21 All of the resulting frequency data sets can be found in the supporting information, and mean absolute deviations between the all orbital and RVS/RES frequencies have been abbreviated to MAD.



Results & Discussion Analytical Hessian Calculations Analytical geometric first and second derivatives are available for TDDFT/TDA in Q-Chem, having been reported in the literature by Liu and Liang.19 It is therefore possible to construct the Hessian matrices used in the harmonic vibrational analysis either analytically, or from finite difference calculations. These finite difference calculations can then be done using either analytical gradients of the excitation energies or the excitation energies themselves. While the lower order derivative methods are significantly more computationally expensive than an analytical Hessian calculation, evaluations using energy, gradient, and analytical Hessians have initially been considered for the two diatomic systems CO and CN-, shown in Figure 1.


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Figure 1. Deviations from the frequencies calculated with an untruncated virtual space for the different levels of analytical derivatives available for CN- (top) and CO (bottom).

The calculated values are identical for the three methods within numerical error for the full calculation. However, the frequencies calculated for the S1 states of these two molecules are found to degrade at a slower rate when reducing the virtual orbital space for the Hessian matrices calculated using energy finite differences, compared to either the gradient or direct Hessian evaluations. When using excitation energies, the virtual space of CN- can be reduced by 60%, maintaining an error of < 1 cm-1 and by up to 85 % for an error of < 2 cm-1. In contrast the gradient and Hessian based analytical approaches already differ from the full result by 1.19 and 5.84 cm-1 following a reduction of just 10% of the virtual space, with errors of 0.01 cm-1 following a 5 % reduction. For carbon monoxide the S1 state is also more sensitive to the size of the virtual space in general, and only 25 or 45 % of the virtual space can be removed to remain with 1 or 2 cm-1 of the full value, respectively, when using the energy only scheme. Page | 7 ACS Paragon Plus Environment


Again, much larger errors are seen when using analytical gradients or Hessians, with a 5 % virtual orbital reduction tolerated when using analytical Hessians and a 20 % reduction tolerated for gradient finite differences to remain below 1 cm-1 from the unrestricted value. The differences in behaviour seen for the differing levels of analytical derivative are likely either due to the assumptions used in the implementation of the derivative schemes, which have been described in detail elsewhere,19 or in the interface between the derivative codes and the RVS scheme in Q-Chem. As a result, the rest of the frequency calculations and geometry optimizations used in this work will make use of energy only finite difference schemes in order to provide more general insight into the importance that the virtual orbital excitation space has when calculating vibrational frequencies.

Singlet State Frequencies Optimizing the S1 state geometry of hypofluorous acid (HOF) causes population of a σ* antibonding orbital between the oxygen and fluorine atoms, with a subsequent increase in the O-F bond length to 144 pm, from 186 pm in the ground state and a change in the HOF angle from 98.70 to 83.71°. Unlike the diatomic molecules, the introduction of more than two atoms into the system creates the possibility for bending coordinate motions. ‘Floppy’ modes with bending or tortional character are liable to suffer from larger errors. Low frequency floppy vibrational modes tend to be significantly more delocalized, and involve contributions from a wider range of atoms and bonds than higher frequency modes. These modes are therefore likely to be more sensitive to the overall description of the electronic state. Low frequency modes also commonly involve bending and tortional motions with a less steep PES curvature, making them more sensitive to small absolute changes in the PES as a function of geometry. The low frequency ν1 and ν2 modes of HOF increase by 4.37 and 2.48 cm-1, respectively, following removal of 25 Page | 8 ACS Paragon Plus Environment

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% of the virtual orbitals. By contrast, a 50 % and 95 % reduction in the virtual orbitals is needed to produce an error > 1 cm-1 and > 2 cm-1, respectively, for the ν3 C-H stretching mode.

The S1 state of H2CS has nπ* character involving a donor HOMO located primarily on the sulphur atom and an acceptor LUMO with an anti-bonding phase combination between the carbon and sulphur atoms. The ν1 vibration of H2CS can be characterized primarily as a wagging mode with the motion localized on the sulphur atom and one of the hydrogen atoms. This mode has an error of < 1 cm-1 with 40 % of the virtual orbitals frozen, compared to 35 % for the ν2 mode. However, larger errors are seen in this mode at higher levels of orbital reduction, as well as for the corresponding triplet state, which will be discussed in more detail later. Overall, the 1nπ* H2CS state shows some of the lowest errors of the states tested here, with no errors greater than 2 cm-1 found in any of the modes ν2 to ν6 for any of the cutoff values tested (shown in Figure 2), including when 95 % of the virtual orbitals are removed.



Figure 2. Deviations of the restricted space S1 state vibrational frequencies from their unrestricted calculations, shown for the H2CS molecule.

The S1 state of ethene was optimized from a twisted starting structure with the CH2 groups rotated by 90° with respect to one another rather than from the twisted-pyramidalized structure reported in the literature, in order to preserve a higher degree of symmetry and avoid issues with geometry convergence.44 The S1 state for the ground state geometry involves a standard ππ* type transition, however, the HOMO and LUMO orbitals change significantly following this rotation. Up to 10 % of the virtual orbitals can be removed while still giving frequencies within a wavenumber of the full result for all 12 vibrational modes. At a 15 % reduction in the virtual space the four C-H stretching modes above 3000 cm-1 show errors of ca. 1-2 cm-1 from the full result, and with a 20 % reduction several of the low frequency modes below 1000 cmPage | 10 ACS Paragon Plus Environment

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1

also have errors of over a wavenumber. Once 25 % of the orbitals are removed, errors of up

to ca. 31 cm-1 are introduced into the C-H stretching modes, with a 54 cm-1 error found in the ν6 mode and a MAD of ca. 11 cm-1 across all modes. The MAD values are shown in Figure 3. These large MAD errors go away once 30 % of the virtual orbitals are removed and the MAD drops back down to < 1 cm-1. A more detailed orbital-by-orbital reduction of the excitation space shows that these large errors are introduced on removing the LUMO+47 and persist until removal of the LUMO+44, shown in Figure 4. This result suggests a cancellation of errors resulting from the removal of these orbitals.

Figure 3. Mean absolute deviations of the restricted space S1 state vibrational frequencies from their unrestricted calculations, shown for the ethene molecule. A negative consequence of the relatively large errors see for ethene, together with the increased vibrational degrees of freedom compared to the other species tested, is that with 40 % or more of the virtual space removed, the ν6 and ν7 normal modes undergo energy reordering with respect to the full calculation, as do modes ν11 and ν12, and errors of several wavenumbers



persist at the higher levels of reduction. Similar spikes in the errors of specific modes arising when certain orbitals are removed as the description of the S1 state deteriorates further.

Figure 4. The virtual orbitals of ethene being removed in the 20-25 % (top line) and 25-30 % (bottom line) reductions of the virtual orbital space.

Although larger molecules like ethene have more virtual orbitals for a given basis set, and therefore have more scope for reduction in the calculation size, they can also have more complex electronic and vibrational structures, and the increased degrees of freedom also increase the likelihood that reordering will occur in the vibrational mode, which can hinder the assignment of experimental data.

Triplet State Frequencies


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The lowest triplet excited states of molecules are also of significant chemical interest as they can remain populated for relatively long periods of time following photo excitation. The T1 states of the two diatomic molecules, CN- and CO, are found to be somewhat less sensitive to reductions in the excitation space than their singlet counterparts, and these two states tolerate a 75 and 45 % reduction in their virtual orbitals, respectively, before the calculation errors rise above a wavenumber. These reductions to the T1 states are significantly larger than the 60 and 25 % reductions tolerated by the S1 states in order to be within a wavenumber from the full calculations. Both sets of values are shown in Figure 5. In order to remain within 2 cm-1 of the full result, both the singlet and triplet states can only be reduced by 85 % in the case of CN-, and by 45 % in the case of CO, at which point both the singlet and triplet states experience a relatively large increase in error, indicating that these orbitals are particularly important for describing the excited states accurately.



Figure 5. Deviations of the restricted space calculated S1 and T1 state vibrational frequencies of CN- (top) and CO (bottom) from their unrestricted calculations.

Similarly, the T1 state of HOF can withstand the removal of 30 % of the virtual orbitals before introducing a 1 cm-1 error, compared with 20 % for the S1 state. Although the errors for the HOF modes are more erratic for the T1 state, a generally lower error profile is observed when removing the virtual orbitals (shown in Figure 6). The one notable exception to this is that the error in the lowest frequency mode jumps to -16.27 cm-1 once 65 % of the orbitals are removed and then falls back to +0.73 cm-1 once 70 % are removed. As with the S1 calculation, the lower two modes with frequencies below 1000 cm-1 both show larger errors sooner compared to the C-H stretching mode, which can withstand a reduction of 40 % with less than 1 cm-1 of error, and up to 95 % reduction without the error rising above 2 cm-1, before it increases sharply from +0.90 to +17.02 cm-1 following the 95 % reduction increment.


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Figure 6. Deviations of the restricted space calculated S1 (top) and T1 (bottom) state vibrational frequencies from their unrestricted calculations, shown for the HOF molecule.

When optimizing the T1 state of H2CS, the gradient remained within ca. 1×10-5 a.u., which may cause additional errors to enter into the calculations, particularly for the lower frequency modes compared to the 1×10-6 a.u. threshold reached for the S1 calculations. The calculation with a 40 % orbital reduction also experienced oscillatory behaviour in the geometry optimization process and failed to converge. The ν1 mode of the H2CS triplet state has a lower initial frequency of 50 cm-1 compared to ca. 353 cm-1 for the S1 state, and there is a > 1 cm-1 error in the ν1 mode following the initial 5 % reduction in the orbitals. The ν5 and ν6 C-H stretching modes also show errors increasing from < 1 cm-1 to > 2 cm-1 following the 20 % virtual space reduction.



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The cutoff values identified for each molecule and state are summarized in Table 1, and highlight the trends previously observed, that the triplet states tend to be less sensitive to the RVS than the singlet states, and that the higher frequency vibrational modes with CH stretching character tend to be less sensitive than the modes with lower frequencies. Table 1. The virtual orbital percentage cutoff values which still produce less than the specified error for the TDDFT states tested.

CO

S1 T1

All Mode 1 cm-1 Cutoff (%) 25 45

CN-

S1 T1

60 75

85 85

-

-

HOF

S1 T1

20 30

20 55

45 45

90 90

H2CSa

S1 T1

35 45

95 45

95 45

95 70

S1 10 10 10 T1 10 15 15 aThe ν mode has been excluded from the analysis of this molecule. 1

10 15

Molecule State

All Mode 2 cm-1 Cutoff (%) 45 45

CH Mode 1 cm-1 CH Mode 2 cm-1 Cutoff (%) Cutoff (%) -

C2H4

Different Basis Sets and Criteria In order to see if these findings hold true for a wider range of basis sets, orbital truncation of the much larger triple-ζ Dunning aug-cc-pVTZ basis set was also tested for the S1 states of CO, CN-, HOF, H2CS, and C2H4, for which aug-cc-pVTZ has 92, 92, 115,142, and 184 basis functions, respectively, compared to 44, 44, 50, 64, and 68 basis functions for 6-311+G(d,p). While percentage based cutoff values have been used successfully in prior RVS studies,31-32, 45 the changes in the total number of virtual orbitals seen when using both different basis sets and when studying different molecules, can limit the generality of percentage based criteria and Page | 16 ACS Paragon Plus Environment

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can lead to excessively conservative cutoff values when used without a priori knowledge of how the specific system will respond. An alternative scheme is to use an orbital energy cutoff value, where orbitals above a certain energy threshold are removed. One way in which this can be further normalized over a range of basis sets and molecules is the normalized virtual orbital threshold (NVT) approach that has recently been proposed for the RVS when using the algebraic diagrammatic construction scheme for the polarization propagator (ADC) to calculate electronic excitation energies.45 In the NVT scheme, the orbital energies are divided by the HOMO to LUMO orbital energy gap, and this approach has previously been found to normalize differences between basis sets, though different thresholds were still found for differing types of excited state.45 The orbital energies used in this scheme have been taken from the ground state reference calculations for the molecules in their all-orbital TDDFT S1 state geometries. The results for the aug-cc-pVTZ basis set in Table 2 indicate that the larger number of virtual orbitals available in the full calculation at this basis set size doesn’t always allow for a larger relative percentage reduction in the virtual orbital space. However, despite the size of the virtual space being between double and triple the size of the 6-311+G(d,p) basis set, the subwavenumber percentage cutoffs for aug-cc-pVTZ differ by at most +/-5 % (ie. the minimum increment tested) for all of the molecules except CN-, which has already been found to be particularly insensitive to RVS reductions. By contrast, the orbital energy based criteria do not appear to be appropriate for RVS-TDDFT vibrational frequencies, with the differences in the NVT cutoffs between the two basis sets ranging from 37.210 for C2H4, to 1.102 for CO. As such, percentage based cutoffs appear to be the most applicable of the tested criteria currently available for RVS frequency calculations, despite their relatively low recommended values.



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Table 2. The percentage cutoffs and corresponding highest energy and NVT virtual orbital cutoffs that produce less than wavenumber error in all modes for the singlet TDDFT states tested. Molecule CO

Basis Set 6-311+G(d,p) aug-cc-pVTZ

Percentage 25 30

Energy (a.u.) 2.756 3.125

NVT 8.694 9.796

CN-

6-311+G(d,p) aug-cc-pVTZ

60 90

1.001 0.314

4.206 1.454

HOF


20 25

3.909

28.743

5.696

40.397

H2CS


35 30

1.660 2.255

13.071 17.481

C2H4


10 5

2.943

40.875

5.544

78.085

Each molecule and basis set combination also has a small number of very high energy virtual orbitals that are energetically separated from the rest of the virtual space by multiple Hartrees. These high-lying virtual orbitals tend to involve a significant degree of s-function character and are observed to be larger in number for basis sets with additional core functions. The core 1s functions of the 6-311+G(d,p) basis set normally consist of a single basis function for each heavy atom, which is composed of six contracted Gaussian functions. In order to test the effects of removing these high-lying orbitals, calculations were performed with these orbitals excluded from the TDDFT calculation using the 6-311+G(d,p) and aug-cc-pVTZ basis set, and a third set of calculations was performed using the u6-311+G(d,p) basis set, where the core 1s functions of the 6-311+G(d,p) set are uncontracted and the six Gaussian functions are allowed to vary independently. The results of these calculations (in Table 3) show that the errors introduced by removing these orbitals are negligible, even when the number of orbitals being removed became relatively large, such as seen for H2CS, where 14 orbitals were removed.


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Table 3. The mean and maximum absolute errors introduced following the removal of the number of highest orbitals indicated (given in cm-1) for each basis set. Molecule CO

Basis Set aug-cc-pVTZ 6-311+G(d,p) u6-311+G(d,p)

# Orbitals 2 2 10

MAD 0.00 0.01 0.00

Max 0.00 0.01 0.00

CN-

aug-cc-pVTZ 6-311+G(d,p) u6-311+G(d,p)

2 2 10

0.00 0.01 0.00

0.00 0.01 0.00

HOF


2 2 12

0.01 0.00 0.00

0.01 0.00 0.00

H2CS


1 1 14

0.01 0.00 0.02

0.07 0.01 0.05

C2H4


2 2 10

0.01 0.03

0.05 0.07

0.00

0.02

Core Orbitals Previous studies that have looked at the effects of reducing the excitation space within TDDFT for other properties and have found that removing excitations from occupied core donor orbitals also be an effective way of reducing the calculation size without introducing significant errors.31-32 However, one recent study found that core basis functions can be important for describing vibrational frequencies in the ground state.46 The effects that removing the core orbitals have on the S1 calculated vibrational frequencies has therefore been tested for these molecules as well, by removing either one or both of the core orbitals in each molecule, as shown in Table 4 to Table 8. In the case of ethene the core orbitals are not localized onto one specific carbon atom but rather involve an in-phase (ip) and out-of-phase (oop) combination of the C(1s) atomic orbitals shown in Figure 7.



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Table 4. The TDDFT S1 frequency of CO and the errors introduced following the removal of the core orbitals indicated (given in cm-1). None 1507.45

O(1s) +0.03

C(1s) -0.87

C(1s)O(1s) -0.84

Table 5. The TDDFT S1 frequency of CN- and the errors introduced following the removal of the core orbitals indicated (given in cm-1). None 2132.11

N(1s) +0.02

C(1s) +0.01

C(1s)N(1s) +0.03

The C(1s) orbital is significantly more important for describing the CO vibrational frequency than the O(1s) orbital, which introduces an error of just +0.03 cm-1 compared to -0.87 cm-1. The effect of removing both core 1s orbitals is also additive to within two decimal places, and causes an overall error of -0.84 cm-1. The errors introduced for CN- are comparatively small, with an error of +0.02 cm-1 when removing for the N(1s) orbital and +0.01 cm-1 when removing the C(1s) orbital. In both cases the error introduced remains less than 1 cm-1. Similarly small errors are seen when removing the O(1s) and F(1s) core orbitals for HOF, though the O-F ν2 stretching frequency is relatively more effected by removal of the F(1s) orbital, with an error of +0.04 cm-1, and for H2CS no errors greater than +/- 0.01 cm-1 are introduced following removal of either the S(1s) or C(1s) orbitals. Table 6. The TDDFT S1 frequencies of HOF and the errors introduced following the removal of the core orbitals indicated (given in cm-1). None 552.52 627.87 3685.18

O(1s) 0.00 -0.01 0.00

F(1s) +0.01 +0.04 0.00


O(1s)F(1s) 0.00 +0.03 0.00

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Table 7. The TDDFT S1 frequencies of H2CS and the errors introduced following the removal of the core orbitals indicated (given in cm-1). None 352.61 809.82 895.50 1377.05 3127.82 3240.15

S(1s) -0.01 0.00 0.00 0.00 0.00 0.00

C(1s) -0.01 +0.01 0.00 0.00 0.00 0.00

C(1s)S(1s) 0.00 +0.01 0.00 0.00 0.00 0.00

Table 8. The TDDFT S1 frequencies of C2H4 and the errors introduced following the removal of the core orbitals indicated (given in cm-1). None 668.18 675.79 892.93 900.56 950.70 1298.70 1300.89 1517.26 2845.94 2846.25 2856.70 2914.49

C(1s) ip +0.10 +0.08 +0.18 +0.12 -0.01 -2.74 0.00 -0.01 -148.43 -0.26 -10.41 +0.04

C(1s) oop +0.08 +0.08 +0.15 +0.14 -0.01 +2.05 -0.01 0.00 +0.06 +0.05 +57.84 +97.94

C(1s)C(1s) +0.20 +0.18 +0.34 +0.27 0.00 -0.06 -0.01 0.00 +0.02 +0.02 0.00 +0.03

Unlike for the other molecules tested, selectively removing either the in-phase or the out-ofphase core orbital from ethene causes very larger errors in several of the vibrational frequencies. The ν6 mode differs from the full result by either -2.74 cm-1 or +2.05 cm-1, depending on which orbital is removed, and an average error of ca. 40 cm-1 is introduced into the C-H stretching modes, with a maximum error of ca. 148 cm-1. These errors fall to a MAD of just 0.09 cm-1 per mode and a maximum error of 0.34 cm-1, when both core orbitals are removed. This result is more consistent with the other molecules of the test set, indicating that it is particularly important to evenly remove delocalized core orbitals that are not symmetrically unique, when applying this variation of the RES.



Figure 7. The occupied core orbitals of ethene in the optimized S1 geometry.

Conclusions The results indicate that vibrational frequencies are more sensitive to the RVS approximation than for vertical excitation energies.31 However, a reduction of ca. 10-20 % of the virtual orbitals is still possible at the 6-311+G(d,p) level while remaining within 1-2 cm-1 of the full results, and larger reductions are possible in specific cases. Furthermore, energy based orbital reduction criteria appear to be less generally applicable than removing a percentage of the virtual orbital space, although the very high energy orbitals with character from the core basis functions can be removed safely. The triplet state frequencies examined were found to be more robust to the removal of virtual orbitals than the equivalent singlet state frequencies, with the exception of 3H2CS. Occupied core orbitals can be removed from the excitation space in most cases without introducing significant error, however, care must be taken when removing a subset of the core orbitals when they are delocalized across multiple atomic centres. Low frequency modes have been shown to be more affected by reductions in the excitation space than higher frequency modes, and more orbitals can be removed for several of the states tested, without introducing large errors into C-H stretching frequencies, though this may not hold true for more complex C-H environments and large relative errors can also sometimes be seen in part due to these modes having larger absolute values. When dealing with larger molecules which possess more degrees of freedom in both the electronic configuration and in terms of the Page | 22 ACS Paragon Plus Environment

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vibrational modes, the picture becomes more complicated, as individual orbitals can become important and start to cause large errors without being obvious a priori. At the same time large molecules increase the possibility that vibrational modes will undergo energetic reordering with the RVS approximation. In these cases a cutoff of no more than 10 % of the virtual orbitals is suggested and additional care should be taken.

ASSOCIATED CONTENT Supporting Information Tables of vibrational frequencies calculated for the molecules and states tested in this study, and images of the HOMO and LUMO orbitals primarily involved in forming the excited states.

ACKNOWLEDGEMENTS I would like to thank the University of Nottingham and the EPSRC for funding (EP/L50502X/1).

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Figure 1. Deviations from the frequencies calculated with an untruncated virtual space for the different levels of analytical derivatives available for CN- (top) and CO (bottom). 210x152mm (150 x 150 DPI)

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Figure 2. Deviations of the restricted space S1 state vibrational frequencies from their unrestricted calculations, shown for the H2CS molecule. 337x258mm (96 x 96 DPI)


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Figure 3. Mean absolute deviations of the restricted space S1 state vibrational frequencies from their unrestricted calculations, shown for the ethene molecule. 189x108mm (150 x 150 DPI)



Figure 4. The virtual orbitals of ethene being removed in the 20-25 % (top line) and 25-30 % (bottom line) reductions of the virtual orbital space. 285x164mm (96 x 96 DPI)


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Figure 5. Deviations of the restricted space calculated S1 and T1 state vibrational frequencies of CN- (top) and CO (bottom) from their unrestricted calculations. 215x165mm (150 x 150 DPI)



Figure 6. Deviations of the restricted space calculated S1 (top) and T1 (bottom) state vibrational frequencies from their unrestricted calculations, shown for the HOF molecule. 337x258mm (96 x 96 DPI)


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Figure 7. The occupied core orbitals of ethene in the optimized S1 geometry. 128x39mm (96 x 96 DPI)



146x82mm (96 x 96 DPI)


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Excited State Vibrational Frequencies: Restricted Virtual Space Time

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