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Parameter: to a function
Configuration: to an application
Argument: to a CLI command
molecular orbitals (MOs), aromaticity
For a 6 annulene:By Polygon Rule: it has floorToOddNumber(6/2) => [3] bonding MOs.By Huckel's Rule: 6 = 4*[1] + 2 => suggests AROMATIC([3] - 1)/2 = [1]
※ remember that annulene is only applicable to the annulenes.
Consider cyclohexane (C6H12):
Originally, each carbon atom has 2 p e-'s - that's 6*2 = 12 pi e-'s in total.According to the Polygon Rule: Its "energy diagram" should be a hexagon, opening 6*2 spots for e-'s to settle in.∵12 pi e-'s = 12 spots in the polygon ∴The polygon should appear "all occupied", including all the bonding and anti-bonding "MOs".We noticed that the cyclohexane has no pi bond. (Does "anti-bonding" mean "canceling the bonding effect of the bonding MOs"?)
Now consider the [6]annulene (a.k.a. "benzene"):
One of the requirements for Huckel's Rule suggests that the molecule should have a continuous cyclic repetition of pattern "C-C=C", which actually reduces 1 e- from each C.So, the total amount of pi e-'s in benzene = 6*2 - 6*1 = 6It's just enough to fill up all the 3 bonding MOs.(We observed 3 pi bonds in the benzene molecule. Does this suggests a "bonding MO" represents a pi bond in the molecule?)
Now consider cyclohexa-1,3-diene (benzene with one C=C bond reduced by H atoms):
2 additional pi e-'s compared to benzene[Polygon Rule] the 2 additional e-'s are placed separately on the 2 degenerate LUMOs in the benzene diagram. Imagine these 2 e-'s are placed together in one of these LUMOs - an anti-aromatic MO is filled, so the molecule structure should have one pi bond less than benzene - and it truly is so.[Huckel's Rule] 8 = 2*4 ===suggests===> anti-aromacitiy <===because that=== we are having 2 MOs with only 1 e- in each (can be proven by Polygon Rule), which makes up a "di-radical" structure.
The bonding/anti-bonding of the MOs - gives the amount of pi bonds in the molecule (just atom-and-atom bonds; not "pi conjugated system"):a filled bonding MO suggest one more pi bond a filled anti-bonding MO suggest one less pi bond The Aromaticity - all about the stability and reactive electrons:anti-aromaticity: has lone e-'s. They are likely to escape/grab e-'s from other molecules (a.k.a. "reactive"), thus making it very hard for this molecule to stay itself (a.k.a. "unstable").aromaticity: has no lone e-'s. Every pi e- are paired in the MOs ,leaving no e-'s wandering about and causing troubles. It's not likely that this molecule change its structure, thus it has a high stability.
There are two main electronic effects that substituents can exert:
RESONANCE effects are those that occur through the p system and can be represented by resonance structures. These can be either electron donating (e.g. -OMe) where p electrons are pushed toward the arene or electron withdrawing (e.g. -C=O) where p electrons are drawn away from the arene.
INDUCTIVE effects are those that occur through the s system due to electronegativity type effects. These too can be either electron donating (e.g. -Me) where s electrons are pushed toward the arene or electron withdrawing (e.g. -CF3, +NR3) where s electrons are drawn away from the arene.
decide the composition of the product mixture of a reaction when:
competing pathways lead to different products
the reaction conditions influence the selectivity
Favors the thermodynamic product - the one with the lower internal energy.
Advantage in competition: product is more stable (due to lower U).
favored in lower temperature (where the Ea barrier cannot be easily overcome).
goes the thermodynamically-controlled pathway
e.g.: 1,4-addition of HBr to Dienes
Favors the kinetic product - the one with the lower kinetic energy.
Advantage in competition: goes faster (due to lower Ea).
favored in high temperature (where the Ea barrier can be easily overcome).
goes the kinetically-controlled pathway
e.g.: 1,2-addition of HBr to Dienes
For rhyming, the "feature vector" is hereinafter referred to as "eigenvector".
The eigenvectors and eigenvalues are used to describe a linear transformation.
The eigenvector describes the direction in which the direction does not change after the linear transformation is applied. The eigenvalues describe how much the vector will be stretched/compressed (as a scaling factor) in the direction above (the one that does not change direction after the linear transformation).
in case
V and W are finite dimensional, and
There are selected bases in these spaces,
Then: all linear transformations from V to W can be represented as matrices.
Under certain conditions (such as a linear transformation whose matrix form is a real symmetric matrix), the eigenvectors and eigenvalues can fully represent a linear transformation.
为了押韵，以下将“特征向量”称作“本征矢”。
本征矢 和 本征值 是用来描述一个线性变换的。
本征矢 描述了：作用该线性变换后，方向不会发生改变的方向。 本征值 描述了：上述（作用该线性变换后，方向不会发生改变的那个）方向上，向量会被拉伸/压缩多少（可以当作缩放系数来看）。
如果
V 和 W 是有限维的，并且
在这些空间中有选择好的基，
则：从 V 到 W 的所有线性变换可以被表示为矩阵。
一定条件下（如其矩阵形式为实对称矩阵的线性变换），本征矢 和 本征值 可以完全表述一个线性变换。
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