# The Eigenproblem Translated for Alignment of Molecules

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results and Discussion

^{10}− 10∙λ

^{8}+ 31∙λ

^{6}− 35∙λ

^{4}+ 11∙λ

^{2}− 1∙λ

^{0}.

^{10}− 313∙λ

^{8}+ 3488∙λ

^{7}− 15456∙λ

^{6}− 34720∙λ

^{5}− 40832∙λ

^{4}− 23808∙λ

^{3}− 5376∙λ

^{2}

_{i,j}= −M

_{j,i}.

^{−15}is actually a “0” and it is necessary to be aware of this type of error coming from “machine epsilon” [16] which is about 10

^{−7}for “single” precision, 10

^{−16}for “double” precision, and about 10

^{−19}for “extended” precision. Most floating-point implementations use “double” precision and thus the listed value (3 × 10

^{−15}) “fits in range”.

_{1}, a

_{2}, and a

_{3}as rotation angles defining the rotation matrices (given below), it is necessary to maximize the variance along the axes of coordinates.

- Since rotation by a
_{0}leaves untouched the “z” coordinate, the first problem is to find a value of a_{0}such that the squared sum of the eigenvalue(s) for the [Dx] matrix is minimized (or its coefficient from Table 9, which is x_{1}·x_{2}= x_{1}·${\overline{x}}_{1}$ = −x_{1}^{2}= −x_{2}^{2}, is maximized); - Next, we need to leave untouched the “x” coordinate—which was already fitted in the first step. For this, we may want to employ rotation by a
_{2}, such that the squared sum of the eigenvalue(s) for the [Dy] matrix is minimized (or its coefficient from Table 9 is maximized); - There is no third step involving the third rotation matrix, because by maximizing (or minimizing) the first two coordinates, we have already employed all coordinates (x and y in the first step; y and z in the second).

_{i}← −x

_{i}” and/or “y

_{i}← −y

_{i}” and/or “z

_{i}← −z

_{i}” transformation will align it.

^{T}and it is anti-symmetric if A = −A

^{T}). On the other hand, the elements of the Cartesian coordinate matrices are mirrored relative to the main diagonal—this property is called reflection symmetry, line symmetry, or mirror symmetry—which makes these matrices very suitable for the same set of operations that are typically employed for symmetric matrices. Further, among the known properties of skew-symmetric matrices is the fact illustrated in Table 8—If A is a real skew-symmetric matrix and λ is a real eigenvalue, then λ = 0, i.e., the nonzero eigenvalues of a skew-symmetric matrix are purely imaginary”. Since a skew-symmetric matrix is similar to its own transposition, they must have the same eigenvalues. It follows that the eigenvalues (λ) of a skew-symmetric matrix always come in pairs (±λ), a property which is also illustrated in Table 8.

## 4. Conclusions

_{2})

^{3}conformational problem.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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x (Å) | y (Å) | z (Å) | Atom and Label | |
---|---|---|---|---|

0.7428 | −1.4498 | −0.0709 | O | 1 |

−1.1425 | 1.1688 | 1.3882 | O | 2 |

1.1461 | 1.0581 | −1.4377 | O | 3 |

−2.754 | −0.3648 | −0.3408 | O | 4 |

2.7344 | −0.2934 | 0.3835 | O | 5 |

−0.7774 | 1.0064 | 0.0187 | C | 6 |

0.7504 | 0.9905 | −0.0675 | C | 7 |

−1.3475 | −0.306 | −0.532 | C | 8 |

1.3187 | −0.2968 | 0.5474 | C | 9 |

−0.671 | −1.513 | 0.1111 | C | 10 |

(1, 9) | (1, 10) | (2, 6) | (3, 7) | (4, 8) | (5, 9) | (6, 7) | (6, 8) | (7, 9) | (8, 10) |

Ad | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Di | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 4 | 3 | 3 | 2 | 3 | 2 | 2 | 1 | 1 |

2 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 2 | 4 | 0 | 3 | 3 | 4 | 1 | 2 | 2 | 3 | 3 |

3 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 3 | 3 | 3 | 0 | 4 | 3 | 2 | 1 | 3 | 2 | 4 |

4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 4 | 3 | 3 | 4 | 0 | 5 | 2 | 3 | 1 | 4 | 2 |

5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 5 | 2 | 4 | 3 | 5 | 0 | 3 | 2 | 4 | 1 | 3 |

6 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 6 | 3 | 1 | 2 | 2 | 3 | 0 | 1 | 1 | 2 | 2 |

7 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 7 | 2 | 2 | 1 | 3 | 2 | 1 | 0 | 2 | 1 | 3 |

8 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 8 | 2 | 2 | 3 | 1 | 4 | 1 | 2 | 0 | 3 | 1 |

9 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 9 | 1 | 3 | 2 | 4 | 1 | 2 | 1 | 3 | 0 | 2 |

10 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 10 | 1 | 3 | 4 | 2 | 3 | 2 | 3 | 1 | 2 | 0 |

3D | 3D Distances | Eigenvalues | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||

1 | 0 | 3.541 | 2.885 | 3.671 | 2.347 | 2.890 | 2.440 | 2.427 | 1.429 | 1.427 | −8.429 |

2 | 3.541 | 0 | 3.638 | 2.817 | 4.264 | 1.427 | 2.395 | 2.430 | 2.985 | 3.008 | −6.218 |

3 | 2.885 | 3.638 | 0 | 4.294 | 2.769 | 2.413 | 1.428 | 2.983 | 2.410 | 3.509 | −2.922 |

4 | 3.671 | 2.817 | 4.294 | 0 | 5.536 | 2.432 | 3.767 | 1.421 | 4.169 | 2.421 | −1.893 |

5 | 2.347 | 4.264 | 2.769 | 5.536 | 0 | 3.762 | 2.406 | 4.183 | 1.425 | 3.627 | −1.275 |

6 | 2.890 | 1.427 | 2.413 | 2.432 | 3.762 | 0 | 1.530 | 1.533 | 2.524 | 2.523 | −1 |

7 | 2.440 | 2.395 | 1.428 | 3.767 | 2.406 | 1.530 | 0 | 2.510 | 1.536 | 2.884 | −0.65 |

8 | 2.427 | 2.430 | 2.983 | 1.421 | 4.183 | 1.533 | 2.510 | 0 | 2.876 | 1.526 | 0 |

9 | 1.429 | 2.985 | 2.410 | 4.169 | 1.425 | 2.524 | 1.536 | 2.876 | 0 | 2.372 | 3.60 × 10^{−15} |

10 | 1.427 | 3.008 | 3.509 | 2.421 | 3.627 | 2.523 | 2.884 | 1.526 | 2.372 | 0 | 22.386 |

Dx | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 0 | 1.8853 | −0.4033 | 3.4968 | −1.9916 | 1.5202 | −0.0076 | 2.0903 | −0.5759 | 1.4138 |

2 | −1.8853 | 0 | −2.2886 | 1.6115 | −3.8769 | −0.3651 | −1.8929 | 0.2050 | −2.4612 | −0.4715 |

3 | 0.4033 | 2.2886 | 0 | 3.9001 | −1.5883 | 1.9235 | 0.3957 | 2.4936 | −0.1726 | 1.8171 |

4 | −3.4968 | −1.6115 | −3.9001 | 0 | −5.4884 | −1.9766 | −3.5044 | −1.4065 | −4.0727 | −2.0830 |

5 | 1.9916 | 3.8769 | 1.5883 | 5.4884 | 0 | 3.5118 | 1.9840 | 4.0819 | 1.4157 | 3.4054 |

6 | −1.5202 | 0.3651 | −1.9235 | 1.9766 | −3.5118 | 0 | −1.5278 | 0.5701 | −2.0961 | −0.1064 |

7 | 0.0076 | 1.8929 | −0.3957 | 3.5044 | −1.9840 | 1.5278 | 0 | 2.0979 | −0.5683 | 1.4214 |

8 | −2.0903 | −0.2050 | −2.4936 | 1.4065 | −4.0819 | −0.5701 | −2.0979 | 0 | −2.6662 | −0.6765 |

9 | 0.5759 | 2.4612 | 0.1726 | 4.0727 | −1.4157 | 2.0961 | 0.5683 | 2.6662 | 0 | 1.9897 |

10 | −1.4138 | 0.4715 | −1.8171 | 2.0830 | −3.4054 | 0.1064 | −1.4214 | 0.6765 | −1.9897 | 0 |

Dy | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 0 | −2.6186 | −2.5079 | −1.0850 | −1.1564 | −2.4562 | −2.4403 | −1.1438 | −1.1530 | 0.0632 |

2 | 2.6186 | 0 | 0.1107 | 1.5336 | 1.4622 | 0.1624 | 0.1783 | 1.4748 | 1.4656 | 2.6818 |

3 | 2.5079 | −0.1107 | 0 | 1.4229 | 1.3515 | 0.0517 | 0.0676 | 1.3641 | 1.3549 | 2.5711 |

4 | 1.0850 | −1.5336 | −1.4229 | 0 | −0.0714 | −1.3712 | −1.3553 | −0.0588 | −0.0680 | 1.1482 |

5 | 1.1564 | −1.4622 | −1.3515 | 0.0714 | 0 | −1.2998 | −1.2839 | 0.0126 | 0.0034 | 1.2196 |

6 | 2.4562 | −0.1624 | −0.0517 | 1.3712 | 1.2998 | 0 | 0.0159 | 1.3124 | 1.3032 | 2.5194 |

7 | 2.4403 | −0.1783 | −0.0676 | 1.3553 | 1.2839 | −0.0159 | 0 | 1.2965 | 1.2873 | 2.5035 |

8 | 1.1438 | −1.4748 | −1.3641 | 0.0588 | −0.0126 | −1.3124 | −1.2965 | 0 | −0.0092 | 1.2070 |

9 | 1.1530 | −1.4656 | −1.3549 | 0.0680 | −0.0034 | −1.3032 | −1.2873 | 0.0092 | 0 | 1.2162 |

10 | −0.0632 | −2.6818 | −2.5711 | −1.1482 | −1.2196 | −2.5194 | −2.5035 | −1.2070 | −1.2162 | 0 |

Dz | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 0 | −1.4591 | 1.3668 | 0.2699 | −0.4544 | −0.0896 | −0.0034 | 0.4611 | −0.6183 | −0.1820 |

2 | 1.4591 | 0 | 2.8259 | 1.7290 | 1.0047 | 1.3695 | 1.4557 | 1.9202 | 0.8408 | 1.2771 |

3 | −1.3668 | −2.8259 | 0 | −1.0969 | −1.8212 | −1.4564 | −1.3702 | −0.9057 | −1.9851 | −1.5488 |

4 | −0.2699 | −1.7290 | 1.0969 | 0 | −0.7243 | −0.3595 | −0.2733 | 0.1912 | −0.8882 | −0.4519 |

5 | 0.4544 | −1.0047 | 1.8212 | 0.7243 | 0 | 0.3648 | 0.4510 | 0.9155 | −0.1639 | 0.2724 |

6 | 0.0896 | −1.3695 | 1.4564 | 0.3595 | −0.3648 | 0 | 0.0862 | 0.5507 | −0.5287 | −0.0924 |

7 | 0.0034 | −1.4557 | 1.3702 | 0.2733 | −0.4510 | −0.0862 | 0 | 0.4645 | −0.6149 | −0.1786 |

8 | −0.4611 | −1.9202 | 0.9057 | −0.1912 | −0.9155 | −0.5507 | −0.4645 | 0 | −1.0794 | −0.6431 |

9 | 0.6183 | −0.8408 | 1.9851 | 0.8882 | 0.1639 | 0.5287 | 0.6149 | 1.0794 | 0 | 0.4363 |

10 | 0.1820 | −1.2771 | 1.5488 | 0.4519 | −0.2724 | 0.0924 | 0.1786 | 0.6431 | −0.4363 | 0 |

x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | x_{7} | x_{8} | x_{9} | x_{10} | |
---|---|---|---|---|---|---|---|---|---|---|

[Ad] | −2.318 | −1.556 | −1.334 | −0.506 | −0.41 | 0.41 | 0.506 | 1.334 | 1.556 | 2.318 |

[Di] | −8.429 | −6.218 | −2.922 | −1.893 | −1.275 | −1 | −0.65 | 0 | 3.60 × 10^{−15} | 22.386 |

[3D] | −8.722 | −5.093 | −3.145 | −2.521 | −1.474 | −1.257 | −1.05 | −0.911 | −0.784 | 25.007 |

[Dx] | 15.299∙i | −15.299∙i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

[Dy] | 9.629∙i | −9.629∙i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

[Dz] | 6.973∙i | −6.973∙i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Matrix (A) | |λ·I − A| Polynomial |
---|---|

[Dx] | λ^{8}∙(λ^{2} + 234.0448052) |

[Dy] | λ^{8}∙(λ^{2} + 92.7157814) |

[Dz] | λ^{8}∙(λ^{2} + 48.6224414) |

**Table 10.**The polynomials of [Dx], [Dy], and [Dz] rotated (15°, 15°, 15°) for CID 444173 (heavy atoms).

Matrix (A) | |λ·I – A| Polynomial |
---|---|

[Dx] | λ^{8}∙(λ^{2} + 162.836846) |

[Dy] | λ^{8}∙(λ^{2} + 90.150945) |

[Dz] | λ^{8}∙(λ^{2} + 47.28921) |

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Jäntschi, L.
The Eigenproblem Translated for Alignment of Molecules. *Symmetry* **2019**, *11*, 1027.
https://doi.org/10.3390/sym11081027

**AMA Style**

Jäntschi L.
The Eigenproblem Translated for Alignment of Molecules. *Symmetry*. 2019; 11(8):1027.
https://doi.org/10.3390/sym11081027

**Chicago/Turabian Style**

Jäntschi, Lorentz.
2019. "The Eigenproblem Translated for Alignment of Molecules" *Symmetry* 11, no. 8: 1027.
https://doi.org/10.3390/sym11081027