JP4827444B2 - A method for experimentally determining electrostatic potential by MEM structural analysis of X-ray diffraction data of crystalline materials - Google Patents
A method for experimentally determining electrostatic potential by MEM structural analysis of X-ray diffraction data of crystalline materials Download PDFInfo
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本発明は、結晶物質のX線回折データをもとに、それらの静電ポテンシャル，電場，電場勾配を実験的に求める方法に関するものである。
即ち、本発明は、上記物質のＸ線回折データに対してMEM構造解析を行って得られる精度の高い電子密度分布から静電ポテンシャルを求めることができるものである。したがって、本発明は、先端材料である強誘電体等の先端材料からたんぱく質等の高分子物質の研究まで、その機能解明のツールとして幅広く利用される。
The present invention relates to a method for experimentally obtaining the electrostatic potential, electric field, and electric field gradient based on X-ray diffraction data of a crystalline substance.
That is, according to the present invention, an electrostatic potential can be obtained from a highly accurate electron density distribution obtained by performing MEM structural analysis on the X-ray diffraction data of the substance. Therefore, the present invention is widely used as a tool for elucidating the functions from advanced materials such as ferroelectrics, which are advanced materials, to research on high-molecular substances such as proteins.
上記ＭＥＭ（最大エントロピー法）構造解析とは、情報理論で発展してきた方法で、何らかの理由で欠損した情報を残っている情報から推定する方法の１つとして種々の分野で用いられている。 The above MEM (Maximum Entropy Method) structural analysis, by methods that have been developed in information theory are used in various fields as one method of estimating the information remaining information lacking any for some reason .
X線回折データをもとに物質の静電ポテンシャルを実験的に求める方法としては，これまで電子密度分布を多重極展開する方法がとられてきた。この方法は平面波の重ね合わせとして与えられた電子密度分布を，原子位置を中心とする球面波の重ね合わせとして展開し，その結果から静電ポテンシャルを計算するものである。 As a method for experimentally determining the electrostatic potential of a substance based on X-ray diffraction data, a method of expanding the electron density distribution to multipole has been used. In this method, the electron density distribution given as a superposition of plane waves is developed as a superposition of spherical waves centered on the atomic position, and the electrostatic potential is calculated from the result.
しかし、この方法では，展開次数をどこまで取るか，また展開で取り込めなかった電子密度をいかに再分配するかなどの任意性があった。また実験的に得られる回折データの数が有限であることに起因する精度の問題もあった。 However, in this method, there is an arbitrary nature such as how much the expansion order is taken and how to redistribute the electron density that could not be taken in by the expansion. There is also a problem of accuracy due to the limited number of diffraction data obtained experimentally.
本発明では、最大エントロピー法（ＭＥM）（非特許文献１，２及び３）を用いることで、これらの問題を解決するものである。
近年、最大エントロピー法（MEM）を用いて上記物質のX線回折データを解析し、その詳細な電子密度分布を求める方法が近年提案され，様々な系に適用されて大きな成功をおさめてきた。最近では並列処理を行うことで，たんぱく質に代表される高分子にも適用が可能となってきている。そこでMEM構造解析により得られる電子密度を静電ポテンシャルの計算に応用することを考えた。上記並列処理とは、複数のＣＰＵを持つコンピュータ又は複数のコンピュータを用いて並列（同時）に処理を行うことで高速なデータ処理を行うことである。 In recent years, a method for analyzing the X-ray diffraction data of the above substances using the maximum entropy method (MEM) and obtaining the detailed electron density distribution has been proposed in recent years and has been successfully applied to various systems. Recently, parallel processing has made it possible to apply to polymers such as proteins. Therefore, we applied the electron density obtained by MEM structural analysis to the calculation of electrostatic potential. The parallel processing is to perform high-speed data processing by performing processing in parallel (simultaneously) using a computer having a plurality of CPUs or a plurality of computers.
本発明は，結晶物質に関して実験的に得られるX線回折データをもとに、その物質の静電ポテンシャルを計算する際に，MEMを用いることで精密な静電ポテンシャルを一意的に求める新しい計算アルゴリズムを提供するものである。 The present invention is a new calculation that uniquely obtains a precise electrostatic potential by using MEM when calculating the electrostatic potential of a crystalline material based on experimentally obtained X-ray diffraction data. An algorithm is provided.
即ち、Ｘ線回折データに対してＭＥＭ構造解析を行うと詳細な電子密度分布を得ることができるので、この得られた電子密度分布を逆フーリエ変換すると観測されていないＸ線回折データを予測することができる。そこで、Ｘ線回折データに対してＭＥＭ構造解析を行うことで得られた電子密度分布と上記観測されていないデータを補完したＸ線回折が存在する場合には、エバルトの方法を用いることで静電ポテンシャルを算出することができる。従来の方法（ＭＥＭ構造解析を用いない方法）では、解析に任意性があり同じデータから異なる静電ポテンシャルが導かれた。本発明においては、解析者の意思によって変化させることができるパラメータが原理的にはなく一意的に求まることになる。 That is, when MEM structural analysis is performed on X-ray diffraction data, a detailed electron density distribution can be obtained. When the obtained electron density distribution is subjected to inverse Fourier transform, X-ray diffraction data that is not observed is predicted. be able to. Therefore, if there is X-ray diffraction that complements the electron density distribution obtained by performing MEM structural analysis on the X-ray diffraction data and the data that has not been observed, it is possible to The electric potential can be calculated. In the conventional method (method not using MEM structure analysis), the analysis is arbitrary and different electrostatic potentials are derived from the same data. In the present invention, the parameter that can be changed according to the intention of the analyst is uniquely obtained instead of the principle.
上記エバルトの方法とは、クーロン相互作用が長距離力であり、その和で表される静電ポテンシャルが実空間（現実の３次元空間）で計算したのでは収束しないので（ほぼ無限に遠くの点からの寄与を積分しなければならない）、クーロン相互作用を上手く計算する方法であり、この方法により、クーロン相互作用を実空間と逆格子空間の両方で計算することで無限和を有限和にすることができる。 The above-mentioned Ewald's method is that the Coulomb interaction is a long-range force, and the electrostatic potential represented by the sum does not converge when calculated in real space (real three-dimensional space) (almost infinitely far away) It is a method to calculate the Coulomb interaction well, by calculating the Coulomb interaction in both real space and reciprocal lattice space. can do.
上記逆格子空間とは、結晶物質のように周期的な構造を持つ物質では、しばしばフーリエ変換を行い、逆格子空間で物事を考えるので、実空間の座標（ｘ，ｙ，ｚ）で表される関数は、フーリエ変換を行うと変数がｘ，ｙ，ｚから逆格子空間の変数（Ｇｘ，Ｇｙ，Ｇｚ）に移ることになる。 The above-mentioned reciprocal space is expressed by the coordinates (x, y, z) in the real space because a material having a periodic structure such as a crystal material often performs Fourier transform and considers things in the reciprocal space. When the Fourier transform is performed, the variable shifts from x, y, z to a variable (Gx, Gy, Gz) in the reciprocal lattice space.
又、本発明の方法を利用することで、結晶物質の詳細な電子密度と対応する静電ポテンシャルを同じアルゴリズムの元で同時に求めることが可能となる。
即ち、従来は、電子密度と静電ポテンシャルは異なる方法で求めていた。しかも、従来の方法では上記のように任意性が存在していた。これに対し、本発明の方法では、電子密度と静電ポテンシャルは同一の方法で求められ、しかも電子密度が決まれば一意的にそして同時に静電ポテンシャルが決まることになる。
In addition, by using the method of the present invention, it is possible to simultaneously obtain the detailed electron density of the crystalline substance and the corresponding electrostatic potential under the same algorithm.
That is, conventionally, the electron density and the electrostatic potential have been obtained by different methods. In addition, the conventional method has arbitraryness as described above. On the other hand, in the method of the present invention, the electron density and the electrostatic potential are obtained by the same method, and when the electron density is determined, the electrostatic potential is uniquely and simultaneously determined.
したがって、本発明は、結晶物質のＸ線回折データに基づいてMEM構造解析を行って最も確からしい電子密度分布を求め，その分布から、測定されていない結晶構造因子を推定し、計算によりその結晶物質の静電ポテンシャルを得、こうして得られる静電ポテンシャルからその結晶物質の電場を計算し、さらにその勾配から電場勾配を得ることにより、結晶物質の原子・分子レベルでの分極の状態を解明する方法である。
即ち、分極した分子のプラスに帯電している部分や、陽イオンは静電ポテンシャルの高いところに引き付けられ、逆に分極した分子のマイナスに帯電している部分や、陰イオンは静電ポテンシャルの低いところに引き付けられるが、そのときに分子やイオンが受ける力の方向を電場が表している。そのため電場が分かれば、分子やイオンの反応経路を予測することが可能になる。又、結晶中の電場勾配は実験的に測定することが出来る場合があり、その場合は、本発明による解析結果を実験結果と比較することが可能となり、結晶の解析に役立つことになる。
Therefore, the present invention obtains the most probable electron density distribution by performing MEM structural analysis based on the X-ray diffraction data of the crystal substance, estimates the crystal structure factor not measured from the distribution, and calculates the crystal by calculation. Obtain the electrostatic potential of a substance, calculate the electric field of the crystalline substance from the electrostatic potential thus obtained, and further obtain the electric field gradient from the gradient, thereby elucidating the state of polarization at the atomic and molecular level of the crystalline substance Is the method.
That is, the positively charged part of the polarized molecule or the cation is attracted to a place with a high electrostatic potential, and the negatively charged part of the polarized molecule or the anion is an electrostatic potential. Although attracted to a low place, the electric field indicates the direction of the force that molecules and ions receive at that time. Therefore, if the electric field is known, it is possible to predict the reaction path of molecules and ions. In some cases, the electric field gradient in the crystal can be experimentally measured. In this case, the analysis result of the present invention can be compared with the experimental result, which is useful for crystal analysis.
強誘電体の構造相転移や，有機物質の光誘起構造相転移，固体表面への原子・分子の結合，分子同士の結合，有機物の反応・触媒作用，たんぱく質の反応や情報伝達など様々な局面で、静電ポテンシャルは重要な役割をはたす。本発明の方法を用いれば，静電ポテンシャルを実験的に決めることが可能となり，原子・分子レベルで分極の状態を詳細に調べることができる。 Various aspects such as structural phase transitions of ferroelectrics, photoinduced structural phase transitions of organic materials, bonding of atoms and molecules to solid surfaces, bonding of molecules, reactions and catalysis of organic substances, protein reactions and information transmission And electrostatic potential plays an important role. If the method of the present invention is used, the electrostatic potential can be determined experimentally, and the state of polarization can be examined in detail at the atomic / molecular level.
即ち、電子密度分布が低いとこころでは静電ポテンシャルは低くなり、高いところでは静電ポテンシャルは高くなる。そのため、電子密度分布に偏り（分極）が生じると（原子核の位置と電子密度分布の重心にずれが生じると）静電ポテンシャルに変化が現れる。分極は電荷密度の積分値で利いて来るので、電荷分布だけを見ていても分極の様子は分かりずらいが、静電ポテンシャルで見るとはっきりと分かることになる。 That is, when the electron density distribution is low, the electrostatic potential is low in the heart, and the electrostatic potential is high at high places. For this reason, when the electron density distribution is biased (polarization) (when there is a shift between the position of the nucleus and the center of gravity of the electron density distribution), a change appears in the electrostatic potential. Since polarization works with the integrated value of charge density, it is difficult to understand the state of polarization even if only the charge distribution is observed, but it can be clearly understood from the electrostatic potential.
また，高分子やたんぱく質においては反応活性化部位を特定し化学反応経路を予測することができる。これらのことから本発明は，新規の材料設計や創薬の分野に貢献するものである。 In addition, in the case of polymers and proteins, it is possible to identify the reaction activation site and predict the chemical reaction pathway. For these reasons, the present invention contributes to the field of novel material design and drug discovery.
即ち、上記反応活性化部位とは、有機物質の多くは分極を起こしており、そのため１つの分子でプラスに電荷した部分とマイナスに電荷した部分が生じており、そこで２つの分子が反応したり結合するときには、お互いに逆の符号に電荷した部分が結合しようとし、ポテンシャルの高い部分は分子のマイナスに電荷した部分と結合しようとすることになる。このように結合しやすい部分が反応活性化部位である。 That is, the reaction activation site is that most of organic substances are polarized, and therefore, one molecule has a positively charged portion and a negatively charged portion, and two molecules react there. When they are bonded, the charged parts are opposite to each other, and the high potential part is connected to the negatively charged part of the molecule. Such a portion that is easily bound is a reaction activation site.
又、上記反応経路とは、反応の際、分子が動いていく経路のことであり、この場合ポテンシャルが分かれば、どのような経路を通って分子が反応活性化部位に動いて行くかを予測できる。 In addition, the above reaction path is a path through which molecules move during the reaction. In this case, if the potential is known, it is predicted what path the molecules will move to the reaction activation site. it can.
図１は実験から得られたSi結晶のX線回折データにおいてMEM構造解析を行い推定された結晶構造因子を実験値と比較したものである。即ち、図１は、MEM構造解析から推定した結晶構造因子（電子密度分布）と実験から得られた結晶構造因子（電子密度分布）の比較を示す図である。この図からMEM構造解析が高次の結晶構造因子まで定量的に実験値を再現していることが分かる。 FIG. 1 shows a comparison of the estimated crystal structure factor with the experimental value by analyzing the MEM structure in the X-ray diffraction data of the Si crystal obtained from the experiment. That is, FIG. 1 is a diagram showing a comparison between a crystal structure factor (electron density distribution) estimated from MEM structure analysis and a crystal structure factor (electron density distribution) obtained from experiments. From this figure, it can be seen that MEM structural analysis quantitatively reproduces experimental values up to higher-order crystal structure factors.
本発明のアルゴリズムでは，まず結晶物質のＸ線回折データに基づいてMEM構造解析を行って最も確からしい電子密度分布を求め，その結果から、測定されていない結晶構造因子を推定し、(2)式によりその結晶物質の静電ポテンシャルを計算する。こうして得られる静電ポテンシャルは，MEM構造解析により得られた同じ結晶物質の電子密度分布に対応したものである。 In the algorithm of the present invention, first, the most probable electron density distribution is obtained by analyzing the MEM structure based on the X-ray diffraction data of the crystal substance, and the crystal structure factor that has not been measured is estimated from the result. (2) The electrostatic potential of the crystalline material is calculated by the formula. The electrostatic potential thus obtained corresponds to the electron density distribution of the same crystalline material obtained by MEM structural analysis.
静電ポテンシャルが得られれば，その勾配を計算することでその結晶物質の電場を計算することが出来る。さらにその勾配を計算することで電場勾配をもとめることが出来る。 If the electrostatic potential is obtained, the electric field of the crystalline material can be calculated by calculating the gradient. Furthermore, the electric field gradient can be obtained by calculating the gradient.
ここでは，強誘電体材料の１つであるPbTiO3に適用した例を示す。図２は、MEM構造解析により得られたPbTiO3の電子密度の(100)面での等高線図を示す図であり、そのMEM構造解析により得られた電子密度の(100)面での分布を青から赤までの色の変化で表したものである。赤いところが電子密度の高いところで，それぞれの原子位置に対応している。図２は、電子密度が赤色、黄色、緑そして青の順序で低下していることを示している（これは、以下の図３及び図４においても同様である。）
一方，図3は、本発明の方法により得られたPbTiO3の静電ポテンシャルの(100)面での等高線図を示す図であり、その(100)面での静電ポテンシャルの強さを青から赤までの色の変化で表したものである。Pb原子が分極して正の電荷が上側に溜まっていることに対応して静電ポテンシャルが高くなっていることが分かる（静電ポテンシャルは電子密度の高いところでポテンシャルが高くなる傾向がある）。
Here, an example applied to PbTiO3, which is one of the ferroelectric materials, is shown. Fig. 2 is a diagram showing the contour map of the electron density of PbTiO3 obtained by MEM structural analysis on the (100) plane. The distribution of the electron density obtained by MEM structural analysis on the (100) plane is shown in blue. It is expressed by the color change from red to red. The red part has a high electron density and corresponds to each atomic position. FIG. 2 shows that the electron density decreases in the order of red, yellow, green, and blue (this is also true in FIGS. 3 and 4 below).
On the other hand, FIG. 3 is a diagram showing a contour map of the electrostatic potential of the PbTiO3 obtained by the method of the present invention on the (100) plane. The strength of the electrostatic potential on the (100) plane is shown from blue. It is expressed by the color change up to red. It can be seen that the electrostatic potential is increased corresponding to the polarization of Pb atoms and the accumulation of positive charges on the upper side (the electrostatic potential tends to increase at higher electron densities).
図４は，PbTiO3の等電子密度面上での静電ポテンシャルの強弱を青から赤までの色の変化で表わした図であり、即ち、PbTiO3の等電子密度面（電子密度が等しい点をつないでできた面）を静電ポテンシャルの強弱により着色したものである。Pb原子に対応する電子密度面は青色から赤色に変化しており，原子が分極を起こしていることが分かる。また，O原子もPb原子ほどではないが分極していることが分かる。一方Ti原子は，色の変化がないので、ほとんど分極していないことが分かる。 FIG. 4 is a diagram showing the strength of the electrostatic potential on the isoelectronic density surface of PbTiO3 by the color change from blue to red, that is, the isoelectronic density surface of PbTiO3 (connecting the points with the same electron density). The surface made of is colored by the strength of the electrostatic potential. The electron density surface corresponding to Pb atoms changes from blue to red, indicating that the atoms are polarized. It can also be seen that O atoms are polarized, though not as much as Pb atoms. On the other hand, Ti atoms have almost no polarization because there is no color change.
本発明により、強誘電体の構造相転移，有機物質の光誘起構造相転移，固体表面への原子・分子の結合，分子同士の結合，有機物の反応・触媒作用，たんぱく質の反応や情報伝達など様々な局面において、X線回折データをもとにして、それらの静電ポテンシャルを実験的に決めることが可能となり，その結果、原子・分子レベルで分極の状態を詳細に調べることができる。また，高分子やたんぱく質においては反応活性化部位を特定し化学反応経路を予測することができる。これらのことから本発明は，新規の材料設計や創薬の分野に貢献するものである。 In accordance with the present invention, structural phase transitions of ferroelectrics, photoinduced structural phase transitions of organic materials, atomic / molecular bonds to solid surfaces, molecular bonds, organic reaction / catalysis, protein reactions and information transmission, etc. In various aspects, it is possible to experimentally determine their electrostatic potential based on X-ray diffraction data, and as a result, the state of polarization can be examined in detail at the atomic and molecular level. In addition, in the case of polymers and proteins, it is possible to identify the reaction activation site and predict the chemical reaction pathway. For these reasons, the present invention contributes to the field of novel material design and drug discovery.
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