Order parameter by instantaneous normal modes

Chapter 2 Instantaneous Normal Mode Theory for Clusters

2.3 Order parameter by instantaneous normal modes

In Sec. 2.2.3, for an cluster of n atoms, I define a quantity of atom j in Eq. (2.41), which is referred as a summation of all vibrational projections, , of atom j. The values of the cluster may be classified according to thepoint group character of the cluster structure at the LES. Thegroup character of the LES is well identified so that the values of all atoms may be split into a number of branches each of which corresponds to a subset of the structure. Moreover, as the structure of a cluster is deformed from the LES by thermal fluctuation, the symmetry of the LES is broken and the group character of the cluster is different from that of the LES. This causes the values of the atoms, which are originally split into a number of branches, to be defiled and then mixed up. As the temperature of the cluster raises to the melting point, the values of all atoms will merge entirely into a small range, within which the value of each atom fluctuates, as a result of the almost destruction of the LES. At this stage, the atoms in the cluster are unable to distinguish by the symmetry character of the lowest energy structure.

In the following, I define an order parameter for a pure cluster of n atoms by the standard deviation of values of all atoms at a temperature T so that the melting phenomenon of the cluster can be characterized by this new order parameter. The order parameter is given as

( ) ( )

where l =1 or 2 and the bar notation in Eq. (2.64) denotes an arithmetic average over atoms in the cluster.

Physically, the order parameter is a quantity measuring t e ordering of a cluster’s structures at a finite temperature relative to its LES. At low temperatures, atoms generally vibrate about the equilibrium positions of the LES so that the value of is of the order of unity. As the structure of a cluster becomes liquid-like at a high temperature, approaches to an extremely small value due to the finite size of the cluster.

19 first averaging over cluster configurations and then over atoms in a cluster. Here a notice should be given: In principle, the of a cluster configuration is calculated with the INM eigenvectors of moment of inertia of a cluster’s configuration wit respecti e to one of its principal axes. By substituting this approximate into Eqs. (2.67) and (2.68), and can be shown explicitly as we are ledto an approximate analytic expression of the order parameter . The calculation using this analytical expression for is certainly numerically straightforward.

It is worth mentioning, furthermore, two advantages in using the approximate : (a) Theoretically, has been reduced to a geometric one which is more directly related to the cluster structure, and (b) the computation of are considerably easier and simplified by the unnecessity of dealing with the Hessian matrix and subsequent diagonalization. Although the approximate is simply related to the cluster structures, the order parameter is nevertheless conceptually originated from the curvatures of potential energy landscape, which are more fundamental for they contain the dynamic information and determine the structural transition of a cluster.

Chapter 3 Point Group Theory for the Lowest Energy Structure of Clusters

In this chapter, by point group theory, I investigate the symmetric properties of clusters at the lowest energy structure (LES). In the first section, I will introduce the basic concept of point group theory. In Sec. 3.2, the and axial point groups, to which the Ag17Cu2 and Ag14 clusters at the LES, respectively, are illustrated.

3.1 Basic concept of point group theory

In this section, I will introduce the definition of a group and show the multiplication structures of elements in a group by multiplication table. According to the multiplication structures, I classify the elements in a group into some subsets; this classification is helpful for understanding the properties of a group more concretely.

3.1.1 Definition of group

In group theory, we use symbols to denote the elements of group in a wider sense. They may represent a number, matrix, linear operator, or geometrical operations such as the rotation of a rigid body, and the collection of elements must possess the definite group properties:

Any collection of elements has the group property if an associative law of combination is valid under the specified manipulation for any sequence of element and for any ordered pair R and S, there is a unique product, written as RS, which(in some agreed sense) is equivalent to some single element T which is also in the collection.

However, a collection which possesses this group property is not sufficient to form a group and there are some other rules that the collection should obey. I give the complete conditions for a group in a more formal definition.

A collection of elements form a group G, if (a) it possesses the group property (b) it contains a unit element E such that RE=R for all R in G; (c)it contains for every element R an inverse, which may be called R^-1, such that RR^-1=E

One may doubt in condition (b) and (c) whether the right unit or right inverse elements would be the left unit or left inverse? Actually, this answer is yes. It can be proven that the right unit or inverse are also the left unit or inverse. Therefore, this property is implied naturally in condition (b) and (c), the additional statements about this are unnecessary.

3.1.2 Generator, subgroup, coset, class

In general, the order of a group is defined by its number of elements. In the point group theory, we concern mainly with groups of finite order and in this case the properties of a group of order g are conveniently summarized in a multiplication table which sets out systematically the products of all g² pairs of elements. Basically, the multiplication table is the array and an example is shown in Table 3.1.

Table 3.1 The general form of the multiplication table for a group including four elements A, B, C and D

By examining the multiplication table, one can notice that all the elements of a group of order g may be expressed as products among a limited number of elements called generators. Consider a more simple case, if is in , , ,…, ,…are also in G, G will be of infinite order unless for some value of . The set of elements is called a cyclic group of order n. Here I have adopted the usual algebraic terminology and write . T e usual “laws of indices” are then valid and note that .

The definition of a group by means of its generators is an exceedingly useful devise. The generator provides a simple method for classifying the symmetry of groups, and this is particularly useful for example given in Sec. 3.2. Therefore, I give a formal definition of generator

A set P of elements of a group G is a system of generators of the group if every element of G can be written as the product of a finite number of factors, each of which is either an element of P or the inverse of such an element.

By the definition of a generator, each element of a group can be written as a product of (positive or negative) integer powers of the generators.

Any collection of the elements of which by themselves form a group is called a subgroup of . Actually, except two kinds of trivial subgroup (or called improper subgroup), the unit element only and the whole group itself, other subgroup are said to be proper subgroups. We should notice that the subgroups of a group do not correspond to a way to partition the group. For this point, we notice that the unit element must be a member of every subgroup.

A useful method to specifically partition a group into distinct sets of elements (no element common to two or more sets) is to introduce the idea of coset. Let and suppose that ,

(i) Every elements of a group appears either in the subgroup or in one of its cosets.

(ii) No element can be common to both a subgroup and one of its cosets.

(iii) No element can be common to two different cosets of the same subgroup.

(iv) No coset can contain the same element more than once.

As a result of above properties of cosets, we can completely partition a group into the distinct cosets.

That is, each element of group can be specifically classified into a subgroup or its cosets. It is inferred that

The order h of any subgroup H must be a divisor of the order g of the group .

By the way, it is also possible to define the left cosets. For a finite group, it can be shown that the left cosets and right cosets give exactly the same partition of the group, though the left and right cosets with respect to one particular element are not necessarily identical.

Another way to partition the group is by using the definition of class. Before introducing that, we consider a relation in the following statement

An element is said to be conjugate to with respect to if

We gather elements conjugate to into a collection, as runs through the whole group . We search

the distinct collections via changing the element one by one of group and the distinct collections are called classes. We see at once that for any two elements belong to the same class, they will be conjugate to each other with respect to some element of this group.

3.2 Axial point group

The clusters studied in this thesis include the Ag14 and Ag17Cu2 metallic clusters, whose lowest-energy structures (LES) belong to and axial point group, respectively. In general, the axial point group of a cluster structure usually can be characterized by a n-fold principle axis, with which the cluster rotates through will return to its original structure, and whose number of fold is larger than any other axis. Customarily, this n-fold principal axis is chosen to be the z axis and is depicted in Fig. 3.1, in which the symbol means any other 2-fold axis possesses the 2-fold rotational symmetry and the symbol is a mirror plane that reflects points from one side to the other side and keeps original structure of a cluster. I will explain in detail later by introducing the symmetrical properties of the and axial point groups.

Fig. 3.1 Conventionally, If an object belongs to the axial point group of n-fold rotational symmetry, the n-fold principal axis is fixed to parallel to the z-axis. For every rotation of with respect to principal axis, the object maintains the original structure. The other symbol represents the rotational symmetry of every rotation with respect to 2-fold rotational axis normal to the principal axis and, customarily, the 2-fold axis is set to be the x-axis. Another is the reflection symmetry with respect to a plane which is denoted by the symbol .

3.2.1 axial point group

In this subsection, I illustrate the symmetric properties of axial point group for the lowest energy structure (LES) of metallic cluster Ag14, which is shown in Fig. 3.2. The LES consists of an icosahedron and a floating atom which is located outside it at an equal distance from three atoms that form a triangular facet of the icosahedron. The icosahedron is not perfect but a slightly deformed one.

By removing away the floating atom, the deformed icosahedral geometry can be relaxed into a perfect icosahedron whose surface consists of 20 identical equilateral triangular facets with a bond-length side of 2.689 Å . By comparing the structure of this perfect icosahedron with the deformed one in Ag¹⁴, we find that their major difference lies in the very small expansion of the triangular facet capped on top by the floating atom and the expanded triangular facet is still equilateral but now having a bond-length of 2.705 Å .

Differing from the perfect icosahedrons, which has the point group symmetry, the geometric structure of the LES of Ag¹⁴takes on the point group with the z-axis passing through the central atom inside the icosahedron and the adatom, which are depicted in Fig. 3.2. [88] The 12 atoms at the vertices of the deformed icosahedron can be recognized as four equilateral triangles on four different planes perpendicular to z-axis shown in Fig. 3.2 (b), (c) and (d). In (d), the three atoms of each equilateral triangle have an equal distance from the floating atom. According to the point group, atoms of the LES of Ag14 are classified into six subsets, denoted by different colors as shown in Fig. 3.2.

The central atom and the floating atom are two subsets each of single atom and are denoted hereafter as C and F, respectively. The other 12 atoms are differentiated into four subsets, with each subset containing the three atoms consisting of one equilateral triangle mentioned above. With distances from the F atom at 2.681, 4.579, 5.438, and 6.573 Å , the four subsets of atoms are S1, S2, S3 and S4

representing from the nearest to the farthest distance, respectively (shown in Fig. 3.2(c)). In the following, I will introduce symmetry operations of the axial point group and it can be evidenced that the six subsets of Ag14 are unable to mix by any of these operations.

With the LES of Ag¹⁴ as an example, I define the three-fold rotational symmetry, which is about the z-axis depicted in Fig. 3.3(a). The is used to denote this rotational symmetry operation for every counterclockwise rotation with respect to the principal axis and each power of keeps the rotational invariance of original structure. Apparently, there are three symmetry elements

generated by the generator, and they form a typical cyclic subgroup by the definition given in Sec. 3.1.2. On the other hand, another typical symmetry element is the reflection symmetry with respect to a plane, in which the points are reflected from one side of the plane to the other side and the original structure is kept invariantly (shown in Fig. 3.3(b)). By convention, the refection symmetry about a mirror plane is denoted by . In the LES of metallic cluster Ag14, one can easily find three mirror planes that contain the principal axis and are vertical to the x-y plane and each plane contains one atom in each of the S1, S2, S3 and S4 subsets. Conventionally, is used to denote this kind of mirror planes. So, the one parallel to the x-axis is assigned as , and the other two are

and , with the superscript increasing as in the counterclockwise direction with respect to the principal axis. For a point group with both the symmetry operations with rotational symmetry about a principal axis and with refection planes passing through the principal axis, this group is called the C3v axial point group. Thus, in the axial point group, we use the capital “C” to denote the rotational symmetry about the principal axis and the subscript “3” means that there is a 3-fold rotational symmetry and the subscript “v” denotes the vertical reflection planes that contain the principal axis. Further, we notice that one can generate other by multiplying with powers of , so that can be considered as a new generator of the axial point group. and this is the critical characteristics to distinguish from the point group [89] (which can be generalized to and the axial point group). The detail multiplications of elements in the axial point group are expressed in Table 3.2, in which one can easily check group properties. The subgroups, cosets, classes and generators of the axial point group are given in Table 3.3.

Table 3.2 Multiplication table of axial point group

Table 3.3 The symmetry operators of the axial point group are given in the 2^nd and 3^rd rows. The elements in the 2^st row form a proper subgroup. The elements in the same block belong to a class. Generators are mark by red. Left coset is in the 3^rd row.

axial point group

Fig. 3.2 (a) The lowest energy structure (LES) of metallic cluster Ag14 in 3-dimensional real space.

According to the axial point group, the atoms can be classified into six subsets denoted by different colors. The origin of the coordinate systems is set at the central atom (red). The z axis passes through the central and floating (brown) atoms and the x axis, normal to the z axis, is in the plane that contains the z axis and a green atom. (b), (c) and (d) are the three projected views of the LES. In (b), the LES is projected on to the x-z plane. In (c), the LES is projected on to the y-z plane. The red broken lines indicate the four equilateral triangles that are normal to the z-axis and contain the atoms of the four subsets S1, S2, S3 and S4, which are colored with orange, green, purple and blue, respectively. In (d), the LES is projected on to the x-y plane. The three atoms of each S1, S2, S3 and S4

subsets are in a plane and indeed form an equilateral triangle.

Fig. 3.3 The symmetry operations of the axial point group for Ag¹⁴: (a) the rotational operations with respect to the principal axis, (b) the reflection planes denoted by

, , .

3.2.2 axial point group

Metallic cluster Ag17Cu2 takes on its LES c aracterized by two “center” u atoms locating inside 17 “surface” Ag atoms, with the 19 atoms forming two icosahedra with a partial overlap (see Fig. 3.4), The LES of Ag17Cu2 is consistent with our understanding of a bimetallic cluster (BC) that an atom of smaller size prefers to be surrounded by atoms of larger size (ionic radii of Cu and Ag are 0.96 Å and 1.26 Å , respectively). This mixing tendency arises from the size-mismatched disparity and has previously been noted in the literature for the noble-metal-based BCs [90-[92]. This subsection is concerned with a study for the LES of Ag17Cu2 from the point group theory. In this LES, the 19 atoms of Ag17Cu2 cluster are thus classified into four subsets shown in Fig. 3.4. The four subsets consist of the two centrally located Cu atoms, the top and bottom Ag atoms, the five Ag atoms occupying the middle pentagonal ring, and the ten Ag atoms sitting in the upper and lower pentagonal rings, which I denote hereafter as Cu⁽²⁾, Ag⁽²⁾, Ag⁽⁵⁾, and Ag⁽¹⁰⁾, respectively.

The LES of Ag17Cu2 belongs to the symmetry [88] of point group (shown in Fig. 3.5(a)), which is described in the following: With the principal axis that goes along the two Cu atoms, the LES corresponds to symmetry operation. The axes of 2-fold rotational symmetry are the ones that are perpendicular to the principal axis and pass through one of the Ag atoms in the middle pentagonal ring, which is in the xy-plane. As shown in Fig. 3.5(b) and (c), rotating the atoms in the cluster an angle of with respect to one of the axes of 2-fold rotational symmetry sends the structure of the cluster into

itself. Conventionally, the 2-fold rotational symmetries are denoted as with i=1 to 5: is referred to the one with the x-axis as the rotational axis and the others are indicated with superscript increasing in the counterclockwise direction with respect to the principal axis. Totally, there are five two-fold rotational symmetric operators, which can be generated by multiplying with each of the five operators , so that the can be referred as a new generator of axial point group.

Further, we should consider the reflection symmetry. As shown in Fig. 3.5(b), there are five mirror planes, with each plane containing the principal axis and one 2-fold rotational axis of the operation. We use , with i=1 to 5, to denote the five reflection symmetries with respect to each one of the five mirror planes. Indicated in Fig. 3.5(c), the other reflection plane is the horizontal plane containing the middle pentagonal ring. With respect to this refection place, the atoms above and below it reflect to each other such that the cluster structure maintains and we denote this symmetry operation

在文檔中金屬叢集的熔化行為: 依據瞬間正則模分析的次序參數 (頁 27-0)