Analytic fit

In order to improve the performance of the fit without a large numerical effort, the final potential

V 00 is not directly fitted from the most accurate 1119 grid points.

The procedures were listed in the following and in Figure 3-2:

1. Computing all coefficients in Eqs. (2.9)–(2.14) of 1119 grid points, later did the average procedure as shown in Eqs. (2.15). This step generated a set of energies called

E 00.

2. Computing additional 15495 grid points for ( ,r s at CCSD(T)-F12b/AVTZ _{c c}) level. The set of energies is called E_{F12 /b TZ}. Fit E_{F12 /b TZ} to generate the analytic representation of the potential energy surface called V_{F12 /b TZ}.

3. Selecting the same 1119 grid points from 15495 grid points in { , ,R  _{1 2}, }

30

coordinates. Calculating the difference _{12 /}

00 F b TZ

E E E

   . Fit E to

generate the analytic representation of the potential energy surface called V . 4. The resulting potential energy surface _{12 /}

00 F b TZ

31

1

1 2 , , ,

, , ,

(2 ( 2 cos(

( )

, , ) ) )

, _{i j k n} ^{j n} _i

i j k n

f j f n k

V R d

R

  

  

 



A FORTRAN program computing the fitted surfaces was developed by Michał Slawik^[35] and be used in the Thesis.

32

Figure 3-2 The fitting procedure.

33

Chapter 4

Results and discussions

4.1 Benchmark calculations of interaction energies

In the following sections, the interaction energies at

1 2

{ , ,R  , , , }={6.5, 90, 0, 0, 2.409, 1.449}r s will be discussed with respect to different levels of CC theory and different basis sets in order to find the most suitable theory level and basis set combinations to calculate the interaction energy of H2–HCl complex. The main reason for choosing this geometry is that it is located on the grid points { , ,R  _{1 2}, } close to the interaction energy minimum. The interaction energies changed with varying basis set at this geometries is relatively large than at other geometries’.

4.1.1 CCSD(T) part

The CCSD(T) and CCSD(T)-F12b interaction energies computed in different basis sets with and without midbond functions, are presented in Table 4-1. Table 4-1 displays conventional CCSD(T) results for basis sets AVXZ, where X=D–6 and CCSD(T)-F12b results, with and without the scaling of triples, for basis sets AVXZ (X=D–5).

Results shown that for both the conventional CCSD(T) method and explicitly correlated extension of CCSD(T), obtained with addition of mindbond functions converge much faster than the results computed without midbond functions. The

34

CCSD(T)/CBS(5-6)+M result which derived from two largest AV5Z+M and AV6Z+M basis sets can be considered as the benchmark value and the difference between the value of CCSD(T)/CBS(5-6)+M and CCSD(T)/CBS(Q-5)+M can be considered as the error, then we get the -206.080 0.4 cm^-1 as the benchmark value of the frozen-core CCSD(T)/CBS limit. The result is consistent with the both CCSD(T)-F12b/CBS(Q-5)+M (with and without scaled triple) results. Finally for calculating potential energy surface, use CCSD(T)-F12b/CBS(Q-5)+M approach which gives here a value -205.835 cm^-1 with the estimated error 0.4 cm^-1.

Since the value -205.835 cm^-1 (CCSD(T)-F12b/CBS(Q-5)+M) is within the error range of my benchmark value (-206.080 0.4 ), I can assume that it reproduces the benchmark value and the expected error should be below 0.4 cm^-1. It is very unlikely that the true value for CBS limit of CCSD(T) method is somehow above -206.080 cm^-1 (since the trend for CBS(T-Q)+M, CBS(Q-5)+M, CBS(5-6)+M goes from smaller values to bigger ones).

There is one more possibility to estimate the interaction energy due to fact that the scaled-triples result for the basis set with midbond functions tend to overshoot the interaction energy^[11]. The CCSD(T)-F12b (without scaling) results tend to somewhat underestimate it. Therefore good estimation can be provided by averaged value of CCSD(T)-F12b methods, with and without scaled triples, which gives value -206.118 cm^-1. This value is again within the error range of my best estimations for CCSD(T)/CBS(5-6)+M and CCSD(T)-F12b/CBS(Q-5)+M.

35

Table 4-1 Interaction energies (in cm^-1) calculated by CCSD(T) and CCSD(T)-F12b (both with and without scaled triple excitations) methods for diffent basis sets.

Basis CCSD(T)-F12b CCSD(T)-F12b

CBS(T-Q)+M -207.061 -206.788 -209.320

CBS(Q-5)+M -205.835 1.23 -206.401 0.53 -206.487 2.83

* E(X) stands for the interaction energy calculated in the AVXZ basis set.

36

4.1.2 CCSDT and CCSDT(Q) contributions

Since nowadays inclusion of higher excitations beyond CCSD(T) method are available, I tested the inclusion of full triple excitations CCSDT, of noniterative quadruple excitations CCSDT(Q), and of full quadruple excitations CCSDTQ.

Let’s consider four quantities related to various levels of excitations in the entirely and assumed that only 0.01 cm^-1 for the uncertainty comes from the neglect of full quadruples and all higher coupled-cluster excitations.

The contribution of the midbond functions is very small, as well, especially if we consider the computational times increases strongly with number of functions.

The sum E_int^{( ) ( )QT} converges faster with the basis set than the E_int^{T( )T} and

( )Q T

Eint

 ^ terms separately, but one should exercise a great deal of caution. The estimated CBS limit for all three quantities is as follows: E_int^{T( )T} =-1.430.25 cm^-1 (the CBS(Q-5) result and the estimated error is the difference between CBS(Q-5) and AV5Z), E_int^(Q)T=-2.150.16 cm^-1 (the CBS(T-Q) result and the estimated error is the difference between CBS(T-Q) and AVQZ) and a sum of the above two values

37

(Q) ( )T

Eint

 ^ =-3.580.30 (the estimated error is calculated by 0.25²0.16² ). We can

compare it to another CBS limit E_int^{( ) ( )QT} /CBS(T-Q) value of -3.720.20cm^-1 (the estimated error is the difference between CBS(T-Q) and AVQZ). The two estimates of

( ) ( )Q T

Eint

 ^ are perfectly consistent.

Here I tried to find a better way of calculating the E_int^{( ) ( )QT} correction without a need to compute CCSDT(Q) for any basis larger than AVDZ+M/AVDZ because computational cost is too big for computing it for the whole potential energy surface.

The difference E_int^{T( )T} decreases as the basis set size increases while the

( )Q T

Eint

 ^ difference shows opposite trend. The best approach in this case would be to

calculate E_int^{T( )T} in a larger AVTZ basis set and E_int^{( )QT} in AVDZ. This gives a value of -3.55 cm^-1 which is a very reasonable estimate. The error which comes from different basis sets might cancel each other. I can also try to calculate E_int^{T( )T}

/AVTZ+M and E_int^{( )QT}/AVDZ+M which gives -3.46 cm^-1 but this does not seem like any improvement. Also adding midbond functions to a basis set as small as AVDZ might result in an unbalanced basis that actually performs worse. In the view of the benchmark results with the uncertainty of about 0.3 cm^-1 gives the estimate of the post-CCSD(T) effects to -3.550.30 cm^-1 .

38

Table 4-2 Comparison of the interaction energy for the CCSD(T), CCSDT, CCSDT(Q), and CCSDTQ calculations and values of the E_int^{T( )T} ,

( )Q T

Eint

 ^ , E_int^{( ) ( )QT} , and E_int^{Q( )Q} contributions. All energies are given in cm^-1 .

CCSD(T) CCSDT CCSDT(Q) CCSDTQ

Basis E(X)^* E(X)-E(X-1) E(X) E(X)-E(X-1) E_int^{T( )T} E(X) E(X)-E(X-1) E_int^{( )QT} E_int^{( ) ( )QT} E(X) E_int^{Q( )Q}

AVDZ+M -158.961 -161.971 -3.01 -163.162 -1.19 -4.20 -163.153 0.01

AVTZ+M -193.692 -34.73 -195.966 -34.00 −2.27 -197.792 -34.63 -1.83 -4.10

AVQZ+M -202.361 -8.67 -204.169 -8.20 -1.81 -206.175 -8.38 -2.01 -3.81

AV5Z+M -203.921 -1.56

AV6Z+M -204.683 -0.76 -4.06

CBS(T-Q)M -209.320 -1.47 -2.14 -3.61

CBS(Q-5)M -206.487 2.83

CBS(5-6)M -206.080 0.41

* E(X) stands for the interaction energy calculated in the AVXZ+M basis set.

39

Table 4-2 (continued)

CCSD(T) CCSDT CCSDT(Q) CCSDTQ

Basis E(X)^* E(X)-E(X-1) E(X) E(X)-E(X-1) E_int^{T( )T} E(X) E(X)-E(X-1) E_int^{( )QT} E_int^{( ) ( )QT} E(X) E_int^{Q( )Q}

AVDZ -122.888 -125.611 -2.72 -126.733 -1.12 -3.84 -126.743 -0.01

AVTZ -183.349 -60.46 -185.775 -60.16 -2.43 -187.549 -60.82 -1.77 -4.20

AVQZ -199.442 -16.09 -201.369 -15.59 -1.93 -203.361 -15.81 -1.99 -3.92

AV5Z -202.779 -3.34 -204.462 -3.09 -1.68

AV6Z -204.096 -1.32

CBS(T-Q) -210.647 -212.392 -1.75 -2.15 -3.72

CBS(Q-5) -207.019 3.63 -208.446 3.98 -1.43

* E(X) stands for the interaction energy calculated in the AVXZ basis set.

40

4.1.3 CCSD(T) core electron correction

Here I denote the difference between interaction energy calculated by the all electron-correlated CCSD(T) method and frozen-core electron CCSD(T) method as

AE

Eint

 . Values of the E_int^AE , all electron-correlated CCSD(T), and frozen-core electron CCSD(T) are presented in Table 4-3. Based on the results presented in the table I can draw 3 conclusions:

First, the addition of midbond functions does not make a significant enhancement.

Second, standard AVXZ bases converge very slow.

Third, the agreement between ACVXZ and AWCVXZ results is very good. The latter basis set can be viewed as a little bit better.

The core-valence electron correlation should be more important for interaction energy calculations than the core-core electron correlation, and AWCVXZ emphasizes better description of the core-valence electron correlation. Therefore CCSD(T) with the (AWCVTZ, AWCVQZ) extrapolation gives an excellent estimate -1.80 0.20 cm^-1 for the core electron correlation. The esitmate error comes from the observation that the difference between the CBS(AWCVTZ, AWCVQZ) and CBS(AWCVQZ, AWCV5Z) is 0.08 cm^-1,therefor the best available value is -1.88 0.08 cm^-1. So using value -1.80 the biggest possible error should be smaller than 0.20 cm^-1. It is worth to note that for AWCV5Z basis set the correction is 1.74 cm^-1. All these values are again in the range of estimated error (1.80 0.20 cm^-1). No result in the regular AVXZ bases is even close to this accuracy.

41

Table 4-3 The electron-correlated correction E_int^AE. The energies are given in cm^-1 .

Basis CCSD(T) CCSD(T)-AE

42

4.1.4 Relativistic correction

Here I denote the difference between interaction energy calculated by the all electron-correlated, relativistic corrected CCSD(T) method and all electron-correlated CCSD(T) method as E_int^rel. The relativistic corrections are presented in Table 4-4.

The additional “d” in front of each abbreviated basis set notation means the basis set is decontracted. The relativistic correction is more important for inner-shell electrons, so midbond functions were not included. Most reasonable basis sets give very close results, but this is not true in the case of AVXZ sets. Sufficient flexibility of the basis set in the large-exponent range is needed. Such flexibility is not provided by the AVXZ basis set and that explains here the slow convergence with the size of basis set.

The ACVXZ and AWCVXZ bases are better because they provide additional functions with large exponents, but decontracted basis sets have even more flexibility and should be still better.

Decontracted AVXZ bases are easier to work with than decontracted ACVXZ bases where the convergence problem makes calculation difficult. This problem rises from the additional tight exponents in dACVXZ bases which create nearly linear dependencies. For these reasons, use the dAVQZ basis set which gives the relativistic correction of 1.06 0.05 cm^-1. The uncertainty is assumed somewhat arbitrarily, the data presented in the Table 4-4 suggest that the error should be much smaller.

43

Table 4-4 The relativistic correction (E_int^rel). The energies are given in cm^-1 .

Basis CCSD(T)-AE

(With all electron correlated)

CCSD(T) (With relativistic correction,

all electrons are correlated )

rel

Eint



ACVTZ -183.057 -182.121 0.94

ACVQZ -199.832 -198.916 0.92

ACV5Z -204.188 -203.242 0.95

CBS(T-Q) -211.502 -210.592 0.91

CBS(Q-5) -208.537 -207.615 0.92

dACVTZ -185.016 -183.941 1.07

dACV5Z^* -204.185 -203.131 1.05

AWCVTZ -182.804 -181.841 0.96

AWCVQZ -199.979 -199.058 0.92

AWCV5Z -204.276 -203.330 0.95

CBS(T-Q) -211.669 -210.746 0.92

CBS(Q-5) -208.521 -207.596 0.92

* dACVQZ basis set result did not converge.

44

Table 4-4 (continued)

Basis CCSD(T)-AE

(With all electron correlated)

CCSD(T) (With relativistic correction,

all electrons are correlated )

rel

Eint



AVTZ -183.238 -182.863 0.38

AVQZ -199.412 -199.185 0.23

AV5Z -203.609 -203.119 0.49

AV6Z -205.241 -204.299 0.94

CBS(T-Q) -210.987 -210.674 0.31

CBS(Q-5) -208.918 -208.607 0.31

CBS(5-6) -207.705 -206.974 0.73

dAVTZ -185.311 -184.233 1.08

dAVQZ -199.676 -198.615 1.06

dAV5Z -203.488 -202.433 1.06

CBS(T-Q) -211.152 -210.091 1.06

CBS(Q-5) -208.088 -207.028 1.06

45

4.1.5 Total energy

Adding all the contributions to the interaction energy and calculating error by taking the square root of the sum of the squares of the respective uncertainty, we estimated interaction energy for -210.13 0.54 cm^-1 at the { , ,R   _{1 2}, , , }r s ={6.5, 90, 0, 0,

2.409, 1.449} geometry. A linear addition of uncertainties would give 0.95 cm^-1, but that would be too conservative as different errors act in fairly random way. In the next section, I tried to verify that the theory level and basis set mentioned previously can be applied to calculating the whole potential energy surface by repeating similar calculations of interaction energies at several different geometries.

4.2 Interaction energy at general geometries

The following discussed geometries are all with fixed H2 and HCl bond distance at s=1.449 bohr, r=2.409 bohr, respectively.

4.2.1 {R, θ

₁

, θ

₂

, }={6.5, 90, 180, 0}

CCSD(T) and CCSD(T)-F12b interaction energies and convergence behavior of different basis sets are shown in Table 4-5. CCSD(T)-F12b results show much better

convergence than CCSD(T)’s. CCSD(T)/CBS(5-6)+M and

CCSD(T)-F12b/CBS(Q-5)+M values are perfectly consistent. The estimated error can be chosen from the difference between CCSD(T)-F12b/CBS(Q-5)+M and CCSD(T)-F12b/AV5Z+M, which gives 0.173 cm^-1. The estimated error at this geometries is smaller than the error at {R, θ₁, θ₂, }={6.5, 90, 0, 0} which is 0.4 cm^-1. Here CCSD(T)-F12b/CBS(Q-5)+M gives very accurate result.

The core electron correction E_int^AE varies with AWCVXZ basis set size but it is

46

all together small. Results are presented in Table 4-6. All basis gives close results. The estimated errors will be less than 0.1 cm^-1. Here I choose the CBS(T-Q) value 0.03 cm^-1 if taking computational time into account.

( ) T T

Eint

 ^ , E_int^{( )QT}, and E_int^{( ) ( )QT} values are presented in Table 4-7. The best

estimate can be chosen from the summation of E_int^{( )QT} /CBS(T-Q), -0.97, and

( ) T T

Eint

 ^ /CBS(Q-5), -0.61, which gives -1.58. Taking computational time into account,

the largest basis calculation that can be done are E_int^{T( )T} /AVTZ and E_int^{( )QT}/AVDZ.

It gives -0.93-0.24=-1.35. The estimated error would be below 0.35 cm^-1. This estimate also covers the E_int^{( ) ( )QT} /CBS(T-Q) result (-1.69 cm^-1).

The relativistic correction E_int^rel are presented in Table 4-8. Like core correction results, all dAVXZ sets gives close results. Here I choose the dAVQZ result -1.2. The estimated errors will be less than 0.1 cm^-1.

From the above discussion, I calculate the total interaction energy as a sumation of CCSD(T)-F12b/CBS(Q-5)+M, E_int^AE/CBS(T-Q), E_int^{T( )T} /AVTZ, E_int^{( )QT}/AVDZ, and E_int^rel/dAVQZ. It gives -99.86 cm^-1. The uncertainty is calculated by the square root of the sum of the squares of the respective errors, which gives 0.39 cm^-1.

47

Table 4-5 CCSD(T) and CCSD(T)-F12b interaction energies (in cm^-1) and convergence behavior of different basis sets.

Basis CCSD(T)-F12b CCSD(T)

* E(X) stands for the interaction energy calculated in the AVXZ+M basis set.

48

Table 4-7 The values of E_int^{T( )T} , E_int^{( )QT}, and E_int^{( ) ( )QT} contributions. The energies are given in cm^-1.

CCSD(T) CCSDT CCSDT(Q)

Basis E(X)^* E(X)-E(X-1) E(X) E(X)-E(X-1) E_int^{T( )T} E(X) E(X)-E(X-1) E_int^{( )QT} E_int^{( ) ( )QT}

AVDZ -55.565 -56.773 -1.21 -57.194 -0.42 -1.63

AVTZ -84.382 -28.82 -85.314 -28.54 -0.93 -86.043 -28.8495 -0.73 -1.66

AVQZ -92.851 -8.47 -93.666 -8.35 -0.81 -94.532 -8.4886 -0.87 -1.68

AV5Z -95.307 -2.46 -96.023 -2.36 -0.72

CBS(T-Q) -0.73 -0.97 -1.69

CBS(Q-5) -0.61

* E(X) stands for the interaction energy calculated in the AVXZ basis set.

49

Table 4-8 The relativistic correction E_int^rel. The energies are given in cm^-1.

Basis CCSD(T)-AE

(With all electron correlated)

CCSD(T) (With relativistic correction,

all electrons are crrelated )

rel

Eint



dAVTZ -84.870 -86.112 -1.24

dAVQZ -93.204 -94.405 -1.20

dAV5Z -95.748 -96.938 -1.19

50

4.2.2 {R, θ

₁

, θ

₂

, }={7, 0, 0, 0}

CCSD(T) and CCSD(T)-F12b interaction energies and convergence behavior of different basis sets are shown in Table 4-9. Both CCSD(T)-F12b and CCSD(T) results converge well. CCSD(T)/CBS(5-6)+M and CCSD(T)-F12b/CBS(Q-5)+M values are consistent, and the difference is around 0.12 cm^-1. The estimated uncertainty would be much less than 0.2 cm^-1.

The core electron correction E_int^AE varied with the AWCVXZ basis sets size.

Results are presented in Table 4-10. Again, all basis gives close results. The estimated errors would be less than 0.1 cm^-1.

account, the largest basis calculation that can be done for all grid points are E_int^{T( )T}

/AVTZ and E_int^{( )QT}/AVDZ. It gives -2.66-0.80=-3.46. Comparing these two values give almost identical results. The estimated errors would be less than 0.1 cm^-1.

Therefor the total interaction energy is [CCSD(T)-F12b/CBS(Q-5)+M]+[E_int^AE /CBS(T-Q)]+[ E_int^{T( )T} /AVTZ]+[ E_int^{( )QT} /AVDZ]+[ E_int^rel /dAVQZ]. The total interaction energy at {R, θ₁, θ₂, }={7, 0, 0, 0} is 138.842 cm^-1. The uncertainty is

51

calculated by the square root of the sum of the squares of the respective errors, this gives 0.39 cm^-1.

52

Table 4-9 CCSD(T) and CCSD(T)-F12b interaction energies (in cm^-1).

Basis CCSD(T)-F12b CCSD(T)

* E(X) stands for the interaction energy calculated in the AVXZ+M basis set.

53

Table 4-11 The values of E_int^{T( )T} , E_int^{( )QT}, and E_int^{( ) ( )QT} contributions. The energies are given in cm^-1.

CCSD(T) CCSDT CCSDT(Q)

Basis E(X)^* E(X)-E(X-1) E(X) E(X)-E(X-1) E_int^{T( )T} E(X) E(X)-E(X-1) E_int^{( )QT} E_int^{( ) ( )QT}

AVDZ 163.206 159.969 -3.24 159.172 -0.80 -4.03

AVTZ 150.181 -13.03 147.517 -12.45 -2.66 145.949 -13.22 -1.57 -4.23

AVQZ 144.947 -5.23 142.742 -4.78 -2.21 140.966 -4.98 -1.78 -3.98

AV5Z 143.682 -1.27 141.686 -1.06 -2.00

CBS(T-Q) -1.87 -1.93 -3.80

CBS(Q-5) -1.78

* E(X) stands for the interaction energy calculated in the AVXZ basis set.

54

Table 4-12 The relativistic correction E_int^rel. The energies are given in cm^-1.

Basis CCSD(T)-AE

(With all electron correlated)

CCSD(T) (With relativistic correction, all electrons are correlated )

rel

Eint



dAVTZ 148.911 148.397 -0.51

dAVQZ 144.483 143.958 -0.53

dAV5Z 143.155 142.624 -0.53

55

4.2.3 {R, θ

₁

, θ

₂

, }={7, 90, 90, 90}

CCSD(T) and CCSD(T)-F12b interaction energies and convergence behavior of different basis sets are shown in Table 4-13. Both CCSD(T)-F12b and CCSD(T) results converge. CCSD(T)/CBS(5-6)+M and CCSD(T)-F12b/CBS(Q-5)+M values are consistent, the difference is around 0.1 cm^-1. The estimated uncertainty would be 0.25 cm^-1 approximately.

The core electron correction E_int^AE varied with the AWCVXZ basis set and extrapolated results are presented in Table 4-14. Again, all basis gives close results.

The estimated errors will be less than 0.05 cm^-1.

I cannot perform CCSDT and CCSDT(Q) with large basis set (AVQZ, AV5Z) calculations because of the convergence problem which comes from nonplanar geometries. Here I assume conservative estimate of 0.35 cm^-1 for the uncertainty.

The relativistic correction E_int^rel is presented in Table 4-15. All dAVXZ sets gives almost identical results. The estimated errors will be less than 0.05 cm^-1.

Therefore, the total interaction energy is [CCSD(T)-F12b/CBS(Q-5)+M]+

[E_int^AE/CBS(T-Q)]+[E_int^{T( )T} /AVTZ (-0.91 cm^-1)]+[E_int^{( )QT} /AVDZ (-0.24 cm^-1)]+

[E_int^rel/dAVQZ]. The total interaction energy at {R, θ1, θ2, }={7, 90, 90, 90} is -39.20 cm^-1. The uncertainty is calculated by the square root of the sum of the squares of the respective errors, this gives 0.40 cm^-1.

56

Table 4-13 CCSD(T) and CCSD(T)-F12b interaction energies (in cm^-1).

Basis CCSD(T)-F12b CCSD(T)

* E(X) stands for the interaction energy calculated in the AVXZ+M basis set.

57

Table 4-15 The relativistic correction E_int^rel. The energies are given in cm^-1.

Basis CCSD(T)-AE

(With all electron correlated)

CCSD(T) (With relativistic correction,

all electrons are crrelated )

rel

Eint



dAVTZ -27.074 -27.167 -0.09

dAVQZ -33.417 -33.504 -0.09

dAV5Z -35.812 -35.896 -0.08

58

4.2.4 Summary of the calculations of interaction

energies

Results presented in section 4.1 and 4.2 show that the theory level and basis sets mentioned in section 4.1 can be applied to calculating the interaction energy surface at grid points of the H₂–HCl complex with the total uncertainty of 0.6 cm^-1. The estimated uncertainty at the different levels of theory and the total uncertainty for 4 chosen geometries are summarized in Table 4-16. The theory level and basis sets used to calculate the potential energy surfaces are summarized in Table 4-17.

59

Table 4-16 Estimated uncertainty at the different levels of theory and the total uncertainty for 4 chosen geometries. The energies are given in cm^-1. Total uncertainties are calculated by the square root of the sum of the squares of the respective uncertainty.



R, ,  1 2,



E_int^{CCSD T( )F12b} E_int^AE E_int^{T( )T} E_int^{( )QT} E_int^rel E _int

{6.5,90,0,0} -205.8350.4 -1.80.2 -3.550.3 1.060.05 -210.130.54

{6.5,90,180,0} -97.3440.17 0.030.1 -1.350.35 -1.20.1 -99.860.41

{7,0,0,0} 143.2920.2 -0.460.1 -3.460.35 -0.530.1 138.840.43

{7,90,90,90} -37.6370.25 -0.320.05 -1.150.35 -0.090.05 -39.200.44

60

Table 4-17 The description of the theory level and basis sets used to calculate potential energy surfaces.

( ) 12 ( ) ( )

Interaction energy were calculated by CCSD(T)-F12b, frozen core

AE

Eint

 aug-cc-pwCVTZ+ aug-cc-pwCVQZ+extrapolation Correction including all electron correlation,

( ) ( )

int^{CCSD T}(all electron-correlated ) int^{CCSD T} (frozen core electron)

E E

( )

T T

Eint

 ^ aug-cc-pVTZ Correction including triple excitations in coupled cluster methods,

( )

 ^ aug-cc-pVDZ Correction including quadruple excitations (perturbative) in coupled cluster methods,

( )

 decontracted aug-cc-pVQZ Correction for relativistic effect,

( ) ( )

int^{CCSD T} (relativistic corrected) int^{CCSD T} , all electron-correlated

E E

61

4.3 The levels of theory and basis sets used to

calculate derivatives f

_ij

(X)

We already investigated the levels of theory and basis sets which should be used to calculate the interaction energies to reach the desired accuracy at the reference points.

Now the question is how to calculate the derivatives fij(X). In the following tests, 429 grid points were chosen out of 1119 gird points. Those grids were chosen by combining the following values of R (in bohr): 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9, 10, 11, 12, 15, 20 with the set of 33 unique combinations of the following angles: 0°, 45°, 90° and 135° for ₁; 0°, 45°, 90°, 135° and 180° for ₂; and 0°, 45° and 90° for  . The bond length of H₂ was fixed at s_c-h_s=1.449 bohr. The calculated 4 dimensional potential energy surfaces only include HCl vibration. The vibrationally averaging formula in this case becomes them by CCSD(T)-F12b method and included the core electron correctionE_int^AE, post

62

CCSD(T) contributions E_int^{T( )T} E_int^{( )QT}, and the relativistic correction E_int^rel. Therefore I generated three different vibrationaly averaged interaction energy surfaces to compare them and test the final approximation proposed by Jankowski^[3]: 1. Derivatives f , ₀ f and ₁ f were calculated from the interaction energies ₂

derived from CCSD(T)-F12b method with inclusion of all E_int^AE, E_int^{T( )T} ,

( )Q T

Eint

 ^ and E_int^rel terms. The four-dimensional vibrationaly averaged

interaction energy surface was calculated by the formula presented above. This potential is noted as V^full  V ₀.

2. Derivatives f , ₀ f and ₁ f were computed from the interaction energies, ₂ which derived from CCSD(T)-F12b method plus only core electron correction

AE

Eint

 . The four-dimensional vibrationaly averaged interaction energy surface

was calculated by the formula presented above, later corrections E_int^{T( )T} ,

( )Q T

Eint

 ^ and E_int^rel terms calculated at (rc, sc) reference H2 and HCl distances were added. This potential is noted as

     

by CCSD(T)-F12b method. The four-dimensional vibrationaly averaged interaction energy surface was calculated by the formula presented above and finally all corrections E_int^AE, E_int^{T( )T} , E_int^{( )QT}and E_int^rel terms calculated at

63

The statistical data for the differences between the vibrationaly averaged potential interaction energy surfaces calculated in 3 different approaches described above,

| V^full - V^{F12b AE} | and | V^full - V^F12b |, are presented in Table 4-18. In both cases, the biggest deviations occurred in the short R distances. Absolute values of these differences are slightly bigger but absolute values of interaction energies are very big here, as well. Additionally some discrepancy for short R distances might be due to the convergence problem we met at the highly repulsive region. For the purpose of potential energy surface ( V^full 0), the maximum deviations occurred at {12, 135, 135, 0} and {6, 90, 0, 0}, but they are in the range not much bigger than 0.1 cm^-1. The uncertainty coming from approximations used in calculating the interaction energy is as big as 0.6 cm^-1, therefore errors generated by simplification of the calculations for derivatives (f1, f2) are much smaller.

These small discrepancies indicate that fairly time-consuming calculations at the higher levels of theory and larger basis sets are only needed for (X, r_c) or (X, r_c, s_c) points. One does not need to apply such high levels to calculate derivatives f₁ and f₂ or

f , ij i j 0. Based on this study about one dimensional averaging, I decide to compute the f_ij derivatives from interaction energies obtained from E_int^{CCSD T( )F12b} /CBS(AVQZ, AV5Z), since it produces negligible error.

64

Table 4-18 The statistical data for | V^full - V^{F12b AE} | and | V^full - V^F12b |. The energies are given in cm^-1.

| V ^full - V^{F12b AE} | | V^full - V^F12b |

Average 0.006 0.012

Standard deviation 0.021 0.053

Maximum

Standard deviation 0.013 0.028

Maximum

Standard deviation 0.008 0.012

Maximum

0.090

at {12, 135, 135, 0}

0.104 at{6, 90, 0, 0}

65 418

7 1 2 1

407

12 4 2 1 1 1 1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 50 100 150 200 250 300 350 400 450

Co un t

E

_int (cm^-1)

|<V

^full

>-<V

^F12b+AE

>|

|<V

^full

>-<V

^F12b

>|

Figure 4-1 Histogram for | V ^full( )X - V^{F12b AE} ( )X | and | V^full( )X - V^F12b( )X |.

66

4.4 Features of potential energy surfaces

The benefit of the two-step fitting can be illustrated by Figure 4-2. It shows that the shape of the potential energy surfaces along ₂ is expected to be close to the line shown by CCSD(T)-F12b/AVTZ results. The spline interpolation based on only few

The benefit of the two-step fitting can be illustrated by Figure 4-2. It shows that the shape of the potential energy surfaces along ₂ is expected to be close to the line shown by CCSD(T)-F12b/AVTZ results. The spline interpolation based on only few

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