Chapter 4 Results and Discussion
4.5 Statistical Analysis for Intruder State Removal Techniques
4.5.3 Optimal Values of Shift Techniques
To find the optimal value of the shift parameter for each method, we computed the combined error for all the spectroscopic constants. The combined error of each method is shown in Figure 4.3. The definition of the combined error is a summation of each mean absolute deviation divided by the accuracy units. The accuracy units are chosen as 0.001 Å for Re, 1 cm−1 for ωe, and 0.01 eV for De. For example, the CASPT2/MOLPRO calculations with σ = 0.0 (see Figures 6.72, 6.77, and 6.82), the mean absolute deviations of the spectroscopic constants are 0.0107 Å for Re, 42.6
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Figure 4.3 Combined errors with various values of the shift parameter for different methods.
The optimal value was determined by evaluating of the combined error for each shift technique. In the CASPT2/MOLPRO calculations, the optimal value of the real shift parameter is 1.0 a.u., which shows that the larger shift parameters can reduce the deviations of the spectroscopic constants. We also suggest that the value of the shift parameter should be larger than 0.4 a.u. for eliminating the intruder states. In the MCQDPT/GAMESS calculations, we propose the optimal value of the ISA shift parameter is 0.3 a.u. using the analogous analysis. In the IPEA-CASPT2/MOLCAS, CASPT2/MOLCAS and G1-CASPT2/MOLCAS methods, we uniformly propose the optimal value of the imaginary parameter is 0.6 a.u. due to the similar behavior of these methods. Using larger values of the shift parameter shall increase the combined
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error for all the CASPT2/MOLCAS methods. Comparing with the CASPT2/MOLCAS method, the combined errors of the modified CASPT2/MOLCAS methods already show a good improvement and present almost a constant within the small values of the shift parameter.
The shift techniques cannot only eliminate the intruder states but also reduce the systematic error. We give two following views. First, the variational method in quantum chemistry always shows an overestimation of the exact energy; however, the second-order perturbation theory generally gives us an underestimated energy owing to the over correction energies. Second, if the infinite large value of the shift parameter is used in the MRPT calculations, the correction energies are almost zero.
The MRPT calculations shall present as the CASSCF level. Therefore, employing the shift techniques suitably can balance this over corrections of the second-order perturbation treatment and close to the exact energy. In the past, the customary value of the shift parameter used in MRPT calculations has generally not been larger than 0.4 a.u. for the real shift parameter22,23, 0.05 a.u. for ISA shift parameter11,24, and 0.3 a.u. for imaginary shift parameter25,26. However, our present study reports that the optimal value of each shift parameter is indeed larger than the customary values. On the other hand, employing too small values of the shift parameter probably raise the risk of large error in the MRPT calculations.
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Chapter 5
Conclusions
The shift techniques applicable to the MRPT methods have been critically evaluated by a statistical approach. The spectroscopic constants of 65 studied states were investigated and compared with the experiment. We therefore propose that the optimal value of the shift parameter is 1.0 a.u. for the real shift technique, 0.3 a.u.
for the ISA shift technique, and 0.6 a.u. for the imaginary shift technique in present Thesis. The research results indicate that the employment of the optimal value can diminish the systematic error of the second-order perturbation theory as well as eliminate intruder states. In this study, the idea of transferring the CASSCF wavefunctions between different programs was successfully implemented. The approach could help us to study the different multireference methods in a uniform way. We could further investigate the behavior of shift techniques applicable to MRPT methods through comparing with other multireference methods such as MRCI, NEVPT2 and so on. Because of the fast development of computer resources, MRPT methods can overcome higher computational demands, which may help to expand their applications field in the future. The shift techniques certainly play an important role for the multireference perturbation theory in this context.
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Chapter 6
Auxiliary materials
Figure 6.1 Spectroscopic constants for the X 1Σ state of BH obtained using + different methods with various values of the shift parameter σ.
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Figure 6.2 Spectroscopic constants for the a Π3 state of BH obtained using different methods with various values of the shift parameter σ.
Figure 6.3 Spectroscopic constants for the A Π state of BH obtained using 1 different methods with various values of the shift parameter σ.
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Figure 6.4 Spectroscopic constants for the b 3Σ− state of BH obtained using different methods with various values of the shift parameter σ.
Figure 6.5 Spectroscopic constants for the X Π state of CH obtained using 2 different methods with various values of the shift parameter σ.
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Figure 6.6 Spectroscopic constants for the a 4Σ− state of CH obtained using different methods with various values of the shift parameter σ.
Figure 6.7 Spectroscopic constants for the A ∆ state of CH obtained using 2 different methods with various values of the shift parameter σ.
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Figure 6.8 Spectroscopic constants for the X 3Σ state of NH obtained using − different methods with various values of the shift parameter σ.
Figure 6.9 Spectroscopic constants for the a ∆1 state of NH obtained using different methods with various values of the shift parameter σ.
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Figure 6.10 Spectroscopic constants for the b 1Σ+ state of NH obtained using different methods with various values of the shift parameter σ.
Figure 6.11 Spectroscopic constants for the A Π state of NH obtained using 3 different methods with various values of the shift parameter σ.
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Figure 6.12 Spectroscopic constants for the X Π state of OH obtained using 2 different methods with various values of the shift parameter σ.
Figure 6.13 Spectroscopic constants for the A 2Σ state of OH obtained using + different methods with various values of the shift parameter σ.
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Figure 6.14 Spectroscopic constants for the X 1Σ state of HF obtained using + different methods with various values of the shift parameter σ.
Figure 6.15 Spectroscopic constants for the X 1Σ state of BF obtained using + different methods with various values of the shift parameter σ.
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Figure 6.16 Spectroscopic constants for the A Π state of BF obtained using 1 different methods with various values of the shift parameter σ.
Figure 6.17 Spectroscopic constants for the a Π3 state of BF obtained using different methods with various values of the shift parameter σ.
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Figure 6.18 Spectroscopic constants for the A Π state of CN obtained using 2 different methods with various values of the shift parameter σ.
Figure 6.19 Spectroscopic constants for the X 1Σ state of CO obtained using + different methods with various values of the shift parameter σ.
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Figure 6.20 Spectroscopic constants for the D ∆ state of CO obtained using 1 different methods with various values of the shift parameter σ.
Figure 6.21 Spectroscopic constants for the A Π state of CO obtained using 1 different methods with various values of the shift parameter σ.
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Figure 6.22 Spectroscopic constants for the I 1Σ− state of CO obtained using different methods with various values of the shift parameter σ.
Figure 6.23 Spectroscopic constants for the a' 3Σ+ state of CO obtained using different methods with various values of the shift parameter σ.
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Figure 6.24 Spectroscopic constants for the d ∆3 state of CO obtained using different methods with various values of the shift parameter σ.
Figure 6.25 Spectroscopic constants for the a Π3 state of CO obtained using different methods with various values of the shift parameter σ.
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Figure 6.26 Spectroscopic constants for the e 3Σ− state of CO obtained using different methods with various values of the shift parameter σ.
Figure 6.27 Spectroscopic constants for the 5Π 1( ) state of CO obtained using different methods with various values of the shift parameter σ.
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Figure 6.28 Spectroscopic constants for the X Π state of FO obtained using 2 different methods with various values of the shift parameter σ.
Figure 6.29 Spectroscopic constants for the X Π state of NO obtained using 2 different methods with various values of the shift parameter σ.
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Figure 6.30 Spectroscopic constants for the B Π state of NO obtained using 2 different methods with various values of the shift parameter σ.
Figure 6.31 Spectroscopic constants for the 2Φ 1( ) state of NO obtained using different methods with various values of the shift parameter σ.
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Figure 6.32 Spectroscopic constants for the a Π4 state of NO obtained using different methods with various values of the shift parameter σ.
Figure 6.33 Spectroscopic constants for the b 4Σ− state of NO obtained using different methods with various values of the shift parameter σ.
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Figure 6.34 Spectroscopic constants for the B' ∆ state of NO obtained using 2 different methods with various values of the shift parameter σ.
Figure 6.35 Spectroscopic constants for the X 3Σ state of Bg− 2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.36 Spectroscopic constants for the X 1Σ state of Cg+ 2 obtained using different methods with various values of the shift parameter σ.
Figure 6.37 Spectroscopic constants for the 1∆g 1( ) state of C2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.38 Spectroscopic constants for the 1Σg+ 2( ) state of C2 obtained using different methods with various values of the shift parameter σ.
Figure 6.39 Spectroscopic constants for the a Π3 u state of C2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.40 Spectroscopic constants for the b 3Σ state of Cg− 2 obtained using different methods with various values of the shift parameter σ.
Figure 6.41 Spectroscopic constants for the A Π1 u state of C2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.42 Spectroscopic constants for the c 3Σu+ state of C2 obtained using different methods with various values of the shift parameter σ.
Figure 6.43 Spectroscopic constants for the d Π state of C3 g 2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.44 Spectroscopic constants for the e Π state of C3 g 2 obtained using different methods with various values of the shift parameter σ.
Figure 6.45 Spectroscopic constants for the C Π state of C1 g 2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.46 Spectroscopic constants for the X 1Σ state of Ng+ 2 obtained using different methods with various values of the shift parameter σ.
Figure 6.47 Spectroscopic constants for the A 3Σu+ state of N2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.48 Spectroscopic constants for the W ∆3 u state of N2 obtained using different methods with various values of the shift parameter σ.
Figure 6.49 Spectroscopic constants for the B Π state of N3 g 2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.50 Spectroscopic constants for the B' 3Σu− state of N2 obtained using different methods with various values of the shift parameter σ.
Figure 6.51 Spectroscopic constants for the a' 1Σu− state of N2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.52 Spectroscopic constants for the w ∆1 u state of N2 obtained using different methods with various values of the shift parameter σ.
Figure 6.53 Spectroscopic constants for the a Π state of N1 g 2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.54 Spectroscopic constants for the C Π3 u state of N2 obtained using different methods with various values of the shift parameter σ.
Figure 6.55 Spectroscopic constants for the X 3Σ state of Og− 2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.56 Spectroscopic constants for the a ∆ obtained using different 1 g methods with various values of the shift parameter σ.
Figure 6.57 Spectroscopic constants for the b 1Σ state of Og+ 2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.58 Spectroscopic constants for the c 1Σu− state of O2 obtained using different methods with various values of the shift parameter σ.
Figure 6.59 Spectroscopic constants for the A' ∆3 u state of O2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.60 Spectroscopic constants for the A 3Σu+ state of O2 obtained using different methods with various values of the shift parameter σ.
Figure 6.61 Spectroscopic constants for the X 1Σ state of Fg+ 2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.62 Spectroscopic constants for the 1Σg− 1( ) state of F2 obtained using different methods with various values of the shift parameter σ.
Figure 6.63 Spectroscopic constants for the 1Πg 1( ) state of F2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.64 Spectroscopic constants for the 1Πu 1( ) state of F2 obtained using different methods with various values of the shift parameter σ.
Figure 6.65 Spectroscopic constants for the 3Πu 1( ) state of F2 obtained using different methods with various values of the shift parameter σ.
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Figure 6.66 Distribution of changes in Re induced by changing the intruder state removal parameter within whole studied shift values.
Figure 6.67 Distribution of changes in Re induced by changing the intruder state removal parameter ignoring the region of yellow boxes in Figures 6.1-6.65.
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Figure 6.68 Distribution of changes in ωe induced by changing the intruder state removal parameter within whole studied range of the shift values.
Figure 6.69 Distribution of changes in ωe induced by changing the intruder state removal parameter ignoring the region of yellow boxes in Figures 6.1-6.65.
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Figure 6.70 Distribution of changes in De induced by changing the intruder state removal parameter within whole studied range of the shift values.
Figure 6.71 Distribution of changes in De induced by changing the intruder state removal parameter ignoring the region of yellow boxes in Figures 6.1-6.65.
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Figure 6.72 Mean absolute deviation from the experiment for Re obtained with CASPT2/MOLPRO method.
Figure 6.73 Mean absolute deviation from the experiment for Re obtained with MCQDPT/GAMESS method.
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Figure 6.74 Mean absolute deviation from the experiment for Re obtained with IPEA-CASPT2/MOLCAS method.
Figure 6.75 Mean absolute deviation from the experiment for Re obtained with CASPT2/MOLCAS method.
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Figure 6.76 Mean absolute deviation from the experiment for Re obtained with G1-CASPT2/MOLCAS method.
Figure 6.77 Mean absolute deviation from the experiment for ωe obtained with CASPT2/MOLPRO method.
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Figure 6.78 Mean absolute deviation from the experiment for ωe obtained with MCQDPT/GAMESS method.
Figure 6.79 Mean absolute deviation from the experiment for ωe obtained with IPEA-CASPT2/MOLCAS method.
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Figure 6.80 Mean absolute deviation from the experiment for ωe obtained with CASPT2/MOLCAS method.
Figure 6.81 Mean absolute deviation from the experiment for ωe obtained with G1CASPT2/MOLCAS method.
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Figure 6.82 Mean absolute deviation from the experiment for De obtained with CASPT2/MOLPRO method.
Figure 6.83 Mean absolute deviation from the experiment for De obtained with MCQDPT/GAMESS method.
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Figure 6.84 Mean absolute deviation from the experiment for De obtained with IPEA-CASPT2/MOLCAS method.
Figure 6.85 Mean absolute deviation from the experiment for De obtained with CASPT2/MOLCAS method.
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Figure 6.86 Mean absolute deviation from the experiment for De obtained with G1-CASPT2/MOLCAS method.
Table 1 The low-lying electronic states included in the MRPT calculations and numbers of available experimental data for diatomic molecules.
System Number of states States Number of experimental data
ωe Re De
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Table 2 Experimental data of the low-lying states for diatomic molecules.
System State Experimental data
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Table 3 Conversion of coefficients from spherical harmonic type to Cartesian type.
Orbital type Cartesian typea Spherical harmonic typeb
S S S
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Orbital type Cartesian typea Spherical harmonic typeb
F Fx3 3 1 5 3 atomic orbitals formed from the linear combinations of spherical harmonic functions. The correspondence of each orbital as follow: l = 0 for S-type, l = 1 for P-type, l = 2 for D-type, etc.
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In Appendix A, the main objective is to provide a theoretical interpretation of the experimentally observed infrared spectra (IR) of p-nitroaniline (PNA), dissolved in CD3CN. We are mainly interested in two electronic states of PNA, its singlet ground state, S0 and the lowest excited triplet state, T1. A Simulation of IR spectrum in a liquid solution requires a time-dependent investigation of dynamics of an ensemble of PNA/CD3CN molecules at some finite temperature. Namely, obtaining the IR spectrum consists of running a molecular dynamics trajectory for such an ensemble followed by a Fourier transformation of the resulting dipole moment autocorrelation function. Unfortunately, the development of computational methods does not provide us with a technique, which is feasible and applicable for our systems, in particular for excited states. For this reason, we employed traditional, time-independent methods to
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interpret the IR spectra of the PNA. Here, we used DFT calculation with the B3LYP exchange correlation functional29,30,31 to optimize structures of the PNA and to simulate the IR spectra in a harmonic approximation.
We attempted to provide the IR spectra from gas-phase calculations for PNA and from analogous polarizable continuum model (PCM) calculations32 but the results were unsatisfactory. Fortunately, we found a different line of approach, which is using explicitly solvated models33 to acquire a good agreement with the experimental results for S0 and T1 of PNA.
A.2 Computational Details
Usually, IR spectra of molecules in liquid phase is interpreted by use of the calculation of a bare, gas-phase molecule, assuming that there is no sharp difference between the gas-phase and solvated system spectra. Here, we investigated the singlet ground state of PNA and compared the result with accurate experimental data. After finding a molecular model that was able to reproduce the observed ground-state spectrum of PNA in a satisfactory manner, we have proceeded to interpretation of the triplet-state IR spectrum of PNA.
All the calculations, including geometry optimization, frequency calculations with IR intensity determination, and the PCM calculation with dielectric constant (here ε = 37.5 for CD3CN) were performed using the DFT/B3LYP/cc-pVTZ computational strategy as implemented in the Gaussian 09 program. It was found that
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slight elongation of the optimized C–NH2 bond in the explicitly solvated model leads to substantial improvement in the simulated spectra. The remaining geometrical parameters were reoptimized with keeping the C–NH2 bond distance constant. The following calculations of vibrational frequencies and IR intensities were fund almost sensitive to this procedure. The simulated IR spectra were plotted by MOLDEN program with a half-width of 8 cm−1. Due to the lack of harmonicity, we uniformly scaled the vibrational frequencies by a constant factor of 0.972.
A.3 Results and Discussion
A.3.1 Failure of Simulated Gas-phase and PCM IR Spectra for Singlet Ground State of PNA
The S0 experimental IR spectrum of PNA dissolved in CD3CN (see Figure A.1a) shows four major bands, located around 1320, 1500, 1600, and 1630 cm−1. According to theoretical calculations, these bands are dominated by the symmetric NO2 stretch, antisymmetric NO2 stretch, and two combinations of the NH2 scissoring motion with the phenyl ring stretch, respectively. The most characteristic of these peaks is the symmetric NO2 stretch with a doublet of peaks centered around 1320 cm−1.
The simulated gas-phase B3LYP IR spectrum of S0 PNA (see Figure A.1b) is in apparent contrast with the experimental results. First of all, the spectral window between 1250 and 1350 cm−1 displays two, well-separated strong bands with reversed
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intensity pattern. Other major bands between 1450 and 1650 cm−1 are distinct from the experiment.
Figure A.1 Experimental and computational IR spectra for singlet ground state of PNA. The asterisk symbol indicates the spectra obtained from the PNA+2ACN with the elongation of C−NH2 model.
These unsatisfactory results are probably attributed to ignoring the solvation of PNA by CD3CN. We have taken into account the solvent effects by combing polarizable continuum model (PCM) with DFT calculation. The simulated B3LYP/PCM IR
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spectrum (see Figure A.1c) does not give any improvement over the gas-phase calculations because of the two following reasons. One is that the resulting spectrum still displays only a single peak corresponding to the doublet at 1320 cm−1. The other is that the intensities of observable bands between 1400 and 1650 cm−1 are relatively weak and mismatch the experiment.
A.3.2 Simulated IR Spectra of S
0PNA Using Explicitly Solvated Models
Fortunately, there is a way to consider the solvent effect by employing an explicitly solvated model of PNA, in which two CD3CN molecules (see Figure A.2b) are hydrogen-bonded to the NH2 group. The resulting spectrum (see Figure A.1d) displays very good agreement with the experimental data except for the missing hump in the 1320 cm−1 band. We have considered a multitude of various explicitly solvated models containing one to six CD3CN molecules (see Figure A.2c) attached either to the NH2 group, the NO2 group, or aligned along the ring. It has been found that all the resulting spectra are quite similar with a strong single band between 1300 and 1400 cm−1 and reasonable agreement with experiment in the 1600–1700 cm−1 region.
For explicitly solvated models with four or more solvent molecules, two split bands
For explicitly solvated models with four or more solvent molecules, two split bands