Topological properties of covalent bonds

Electron densities at BCPs of covalent bonds are almost identical for IAM and IAM-HO dynamic model densities. They are similar for INV and MP dynamic model densities, with a largest difference of only 0.1–0.2 electrons/˚A³ for the most polar bonds (the C–O bonds) (Tables 5.3, 5.5, 5.7, 5.9). However, ρ^IAM(BCP) and ρ^IAM−HO(BCP) are substantially lower than ρ^INV(BCP) and ρ^MP(BCP), with an average difference of 0.57, 0.42 and 0.53 e/˚A³ for C–O, C–N and C–C bonds, respectively.

As discussed in Section 5.4.2, MEM density maps are more alike to each other than the corresponding dynamic model densities, indicating the tendency of the MEM to converge to a density independent of the prior. Accordingly, differences be-tweenρ^MEMINV (BCP) andρ^MEMMP (BCP) are smaller than differences betweenρ^INV(BCP) and ρ^MP(BCP). Differences betweenρ^MEMIAM (BCP) on the one hand and ρ^MEMINV (BCP) or ρ^MEMMP (BCP) on the other hand are much smaller than the differences between corresponding dynamic model densities. Nevertheless, sizeable discrepancies remain of average values of 0.22, 0.11 and 0.14 e/˚A³ for C–O, C–N and C–C bonds, respec-tively. They can be attributed to the use of different priors [Eq. (5.1)] rather than wrong reflections phases [Eq. (5.2)]. This feature is demonstrated by two additional MEM calculations for D,L-serine (Tables 5.11 and 5.12). In method 1 the IAM prior has been combined with reflection phases from the INV model, while in method 2 the INV prior has been combined with reflection phases from the IAM model. Method 1 leads to topological properties at BCPs similar to those ofρ^MEMIAM (x). Method 2 leads to topological properties at BCPs similar to those of ρ^MEMINV (x), although reflection phases of the IAM have been used in the constraint [Eq. (5.2)].

Despite nearly equal values of ρ^IAM(BCP) and ρ^IAM−HO(BCP), MEM-density maps obtained with these priors attain quite different values at BCPs (Tables 5.3, 5.5, 5.7, 5.9). Instead,ρ^MEMIAM−HO(BCP) is much closer toρ^MEMINV (BCP). An explanation for this feature lies in the different values of ρ^IAM(x) and ρ^IAM−HO(x) near the local maxima, as it is governed by different ADPs of IAM and IAM-HO (Section 5.4.2).

Used as prior, they apparently force a different convergence of the MEM, and thus indirectly lead to different density values at corresponding BCPs in MEM density maps obtained with these two priors.

The relations between the values at BCPs of the various density maps can be

Table 5.7: Topological properties of covalent bonds of L-Alanine: ρBCP (e/˚A³; first line) and ∇²ρBCP (e/˚A⁵; second line) for eight different density maps.

Dynamic model density MEM density

Bond IAM IAM-HO INV MP IAM IAM-HO INV MP

C1–O1 2.073 2.074 2.656 2.807 2.378 2.434 2.687 2.781 13.91 13.04 -17.82 -27.51 15.77 4.45 -16.81 -25.78 C1–O2 1.999 1.997 2.577 2.649 2.297 2.356 2.628 2.660 9.25 8.21 -20.13 -24.01 13.90 0.75 -20.71 -25.48 C1–C2 1.172 1.166 1.673 1.696 1.498 1.500 1.659 1.686 0.88 0.98 -12.90 -12.51 -8.19 -8.24 -12.67 -13.31 C2–C3 1.200 1.200 1.642 1.611 1.516 1.519 1.679 1.658 0.51 0.53 -11.16 -10.67 -12.28 -11.58 -16.68 -15.90 C2–N 1.380 1.387 1.736 1.614 1.585 1.592 1.688 1.630 2.33 2.25 -10.06 -11.52 -8.90 -8.55 -10.90 -13.33

summarized as follows:

ρ^IAM(BCP)ρ^IAM−HO(BCP)< ρ^INV(BCP)≤ρ^MP(BCP) ρ^MEMIAM (BCP)< ρ^MEMIAM−HO(BCP)≤ρ^MEMINV (BCP)ρ^MEMMP (BCP)

(5.5)

The implication is that, as far as density values at BCPs are concerned, both the IAM-HO and INV priors appear to be of sufficient quality to produce reliable MEM densities.

For Laplacians, ∇²ρ(x), a larger spread of values at BCPs is observed among the density maps (Tables 5.3, 5.5, 5.7, 5.9). These variations appear to depend on the amount of polar character of the bond. For covalent C–C bonds the agreement between ∇²ρ(BCP) in various density maps is even better than for ρ(BCP), now with

∇²ρ^IAM(BCP)∇²ρ^IAM−HO(BCP)>>∇²ρ^INV(BCP)≥ ∇²ρ^MP(BCP)

∇²ρ^MEMIAM (BCP)∇²ρ^MEMIAM−HO(BCP)≥ ∇²ρ^MEMINV (BCP)∇²ρ^MEMMP (BCP)

(5.6)

where the replacement of ”<” in Eq. (5.5) by ”>” in Eq. (5.6) reflects the generally negative values of the Laplacians. Noteworthy is that the positive values of Lapla-cians at BCPs of C–C bonds inρ^IAM(x) andρ^IAM−HO(x) turn into negative values in

5.4. DISCUSSION 89

Table 5.8: Topological properties of hydrogen bonds ofL-Alanine: ρBCP (e/˚A³; first line) and ∇²ρBCP (e/˚A⁵; second line) for eight different density maps.

Dynamic model density MEM density

Bond IAM IAM-HO INV MP IAM IAM-HO INV MP

O2· · ·H3–N 0.328 0.327 0.231 0.258 0.336 0.336 0.296 0.297 2.68 2.66 3.61 3.52 -1.63 -1.45 1.70 1.19 O2· · ·H2–N 0.283 0.281 0.195 0.215 0.292 0.282 0.237 0.235

2.54 2.52 3.19 3.22 0.22 1.16 2.35 2.55

O1· · ·H1–N 0.286 0.282 0.188 0.206 0.253 0.250 0.212 0.216

2.32 2.27 3.05 3.00 1.14 1.28 1.83 2.23

ρ^MEMIAM (x) and ρ^MEMIAM−HO(x) of magnitudes similar to the magnitudes in ρ^MEMINV (x) and ρ^MEMMP (x). These results show that the MEM gives a good description of covalent C–C bonds for all four priors, and that IAM-HO, INV and MP priors lead to density maps of comparable quality at those BCPs. Covalent C–N bonds possess a small polar component. With one exception for Ala-Tyr-Ala_Etoh, which can be attributed, in part, to the lower quality of the data, they obey the same relations as C–C bonds.

An accurate description of C–O bonds is difficult to obtain by multipole refine-ments, as has been noted by several authors (Roversi et al., 1996; Benabicha et al., 2000; Birkedal et al., 2004). The relatively large variations in the magnitudes of Laplacians at BCPs have been attributed to the large variations of magnitudes of second-order derivatives over short distances and possibly less than perfect radial functions, while at the same time moderate variations in the three eigenvalues λi of the Hessian matrix are magnified into large variations of the Laplacian. Specifically, for the carboxylic C–O bonds in the amino acids Mebs et al. (2006) have reported a spread of 20 e/˚A⁵ for values of Laplacians at BCPs of static MP densities. In view of this spread, we find relations between ∇²ρ(BCP) at C–O bonds of dynamic model densities that are similar to the relations obtained for C–C bonds [Eq. (5.6)],

∇²ρ^IAM(BCP)∇²ρ^IAM−HO(BCP)>> ∇²ρ^INV(BCP)≥ ∇²ρ^MP(BCP)

∇²ρ^MEMIAM (BCP)>∇²ρ^MEMIAM−HO(BCP)>∇²ρ^MEMINV (BCP)≥ ∇²ρ^MEMMP (BCP)

(5.7)

The discrepancies between values of∇²ρ(BCP) in MEM density maps are larger for C–O bonds than for C–C bonds [Eq. (5.7)]. Especially,∇²ρ^MEMIAM (BCP) is positive for most bonds, and ∇²ρ^MEMIAM−HO(BCP) attains positive values for several bonds

(Tables 5.3, 5.5, 5.7, 5.9). Based on the fact that the MP model describes chemical bonding better than the IAM does, one can conclude that the large positive values of ∇²ρ^MEMIAM (BCP) probably will not reflect the values of this quantity in ”true”

density maps. On the other hand, large values ofρ(BCP) along with positive values of ∇²ρ(BCP) could indicate the presence of charge-shift bonds (Shaik et al., 2005;

Zhang et al., 2009). Further research will be required for the development of a better understanding of C–O bonds.

Table 5.9: Topological properties of covalent bonds of Ala-Tyr-Ala_Etoh: ρBCP (e/˚A³; first line) and∇²ρBCP (e/˚A⁵; second line) for eight different density maps.

Dynamic model density MEM density

Bond IAM IAM-HO INV MP IAM IAM-HO INV MP

C2-O1 2.095 2.092 2.495 2.711 2.492 2.540 2.574 2.710 18.36 17.29 -8.97 -14.97 26.04 15.26 -1.03 -11.66 C4-O2 2.061 2.058 2.462 2.807 2.529 2.595 2.633 2.860 14.24 13.12 -12.46 -24.86 18.33 5.91 -12.33 -29.37 C6-O3 2.073 2.071 2.609 2.728 2.515 2.567 2.691 2.743 20.87 20.01 -7.73 -11.58 18.66 9.41 -8.29 -12.69 C6-O4 1.950 1.953 2.473 2.594 2.412 2.446 2.518 2.563 9.58 9.16 -16.32 -20.73 11.71 6.68 -11.82 -14.35 C24-O5 1.660 1.669 1.904 1.954 1.852 1.870 1.946 1.961 3.41 3.55 -10.25 -11.12 1.77 0.58 -10.51 -10.17 C31-O6 1.496 1.505 1.656 1.607 1.711 1.737 1.648 1.594 4.34 4.55 -5.30 -3.16 0.37 -1.21 -2.62 0.65 C1-N1 1.365 1.369 1.700 1.665 1.624 1.648 1.684 1.697 2.85 2.83 -8.81 -8.40 -6.40 -7.15 -11.14 -12.66 C2-N2 1.736 1.738 2.203 2.319 2.128 2.166 2.204 2.277 -1.67 -1.81 -19.98 -21.60 -1.82 -7.76 -17.29 -17.92 C3-N2 1.433 1.436 1.732 1.779 1.700 1.724 1.727 1.780 1.14 1.11 -10.38 -12.35 -8.62 -10.10 -13.18 -14.57 C4-N3 1.754 1.753 2.222 2.416 2.227 2.262 2.288 2.408 -1.38 -1.66 -20.09 -24.30 -8.96 -14.38 -20.57 -22.07 C5-N3 1.428 1.430 1.720 1.797 1.740 1.747 1.718 1.769 1.76 1.74 -9.30 -12.99 -10.48 -10.89 -11.04 -12.61 C1-C2 1.175 1.172 1.744 1.692 1.552 1.562 1.756 1.749 0.31 0.37 -12.23 -13.48 -12.38 -13.19 -20.61 -22.01 C1-C7 1.190 1.191 1.613 1.733 1.574 1.583 1.701 1.717 Continued on next page...

5.4. DISCUSSION 91

Table 5.9: Continued

Bond IAM IAM-HO INV MP IAM IAM-HO INV MP

-0.08 -0.06 -11.39 -13.90 -9.36 -8.69 -14.22 -12.77 C3-C4 1.169 1.168 1.742 1.708 1.551 1.561 1.764 1.748 0.88 0.91 -11.46 -12.48 -10.73 -11.22 -17.12 -18.16 C3-C20 1.140 1.137 1.628 1.603 1.452 1.450 1.634 1.573 0.55 0.60 -11.08 -11.57 -2.97 -1.43 -10.76 -9.83 C5-C6 1.155 1.154 1.634 1.715 1.487 1.478 1.662 1.718 -0.01 0.04 -13.51 -14.84 -11.39 -9.81 -15.41 -16.99 C5-C8 1.167 1.163 1.580 1.614 1.522 1.520 1.610 1.594 0.78 0.85 -9.78 -9.99 -5.84 -5.34 -9.70 -8.89 C20-C21 1.216 1.215 1.700 1.636 1.495 1.483 1.656 1.614 -0.05 -0.01 -11.31 -9.61 -5.17 -3.93 -8.83 -8.28 C21-C22 1.433 1.433 1.955 1.999 1.882 1.889 1.982 2.002 -3.04 -3.02 -16.61 -17.94 -14.38 -13.04 -16.00 -16.24 C22-C23 1.451 1.448 1.984 2.011 1.871 1.861 2.021 2.017 -3.20 -3.14 -16.57 -16.92 -14.02 -11.49 -17.10 -15.70 C23-C24 1.439 1.437 2.019 2.029 1.926 1.929 2.066 2.062 -2.74 -2.69 -16.01 -16.00 -15.05 -14.49 -17.80 -16.99 C24-C25 1.440 1.439 1.997 2.021 1.944 1.944 2.034 2.055 -3.24 -3.21 -17.07 -18.06 -18.62 -16.11 -20.11 -20.86 C25-C26 1.446 1.441 1.999 1.984 1.913 1.902 2.007 1.998 -3.13 -3.03 -16.04 -15.02 -16.85 -13.73 -14.03 -13.65 C21-C26 1.434 1.435 1.957 1.995 1.945 1.964 1.989 2.025 -2.55 -2.55 -15.97 -15.40 -16.51 -15.82 -16.72 -16.71 C31-C32 1.256 1.258 1.670 1.680 1.606 1.612 1.688 1.698 0.76 0.80 -11.02 -10.37 -8.18 -8.28 -11.79 -11.13